phd_thesis_azlan
TRANSCRIPT
A DISCRETE COMPUTATIONAL AESTHETICS MODEL FOR A ZERO-SUM PERFECT INFORMATION GAME
MOHAMMED AZLAN BIN MOHAMED IQBAL
THESIS SUBMITTED IN FULFILMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
FACULTY OF COMPUTER SCIENCE
& INFORMATION TECHNOLOGY UNIVERSITY OF MALAYA
KUALA LUMPUR
SEPTEMBER 2008
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ABSTRACT
One of the best examples of a zero-sum perfect information game is chess. Aesthetics is
an important part of it that is greatly appreciated by players. Computers are currently
able to play chess at the grandmaster level thanks to efficient search techniques and
sheer processing power. However, they are not able to tell a beautiful combination from
a bland one. This has left a research gap that, if addressed, would be of benefit to
humans, especially chess players.
The problem is therefore the inability of computers to recognize aesthetics in the game.
Existing models or computational approaches towards aesthetics in chess tend to
conflate beauty with composition convention without taking into account the
significance of the former in real games. These approaches also typically use fixed
values for aesthetic criteria that are rather inadequate given the variety of possibilities
on the board. The goal was therefore to develop a computational model for recognizing
aesthetics in the game in a way that correlates positively with human assessment.
This research began by identifying aesthetics as an independent component applicable
to both domains (i.e. compositions and real games). A common ground of aesthetic
principles was identified based on the relevant chess literature. The available knowledge
on those principles was then formalized as a collection of evaluation functions for
computational purposes based on established chess metrics.
Several experiments comparing compositions and real games showed that the proposed
model was able to identify differences of statistical significance between domains but
not within them. Overall, compositions also scored higher than real games. Based on the
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scope of analysis (i.e. mate-in-3 combinations), any such differences are therefore most
likely aesthetic in nature and suggest that the model can recognize beauty in the game.
Further experimentation showed a positive correlation between the computational
evaluations and those of human chess players. This suggests that the proposed model
not only enables computers to recognize aesthetics in the game but also in a way that
generally concurs with human assessment.
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ACKNOWLEDGEMENTS
I would like to express my appreciation to my supervisor, Prof. Dato’ Ir. Dr. Mashkuri
Hj. Yaacob, who had the foresight to accept me as his doctoral student. I have benefited
from his experience, advice and the intellectual freedom he afforded to me during the
research period.
I would also like to thank John McCarthy (Stanford University, USA) for essentially
suggesting to me what I think is possibly the best approach to this research topic;
Michael Negnevitsky (University of Tasmania, Australia) for the meaningful
discussions we had about my research; Jonathan Levitt (Grandmaster of chess, UK) for
continuously and tirelessly accommodating my questions; David Friedgood (FIDE
Master and International Master of chess solving, UK) for his feedback and willingness
to share his connections to resourceful people; Peter Lamarque (University of York,
UK) who motivated me to improve my writing; and Malcolm McDowell (British Chess
Problem Society) for supplying me with several rare manuscripts on the royal game.
I also want to thank Brian Stephenson (UK) for his collection of chess compositions
which has become integral to this work; Daniel Freeman (Chessgames.com, Florida,
USA) for his support and commitment with regard to my online surveys; and the ICGA
Journal editorial board and reviewers, for their detailed and fruitful comments over the
years on various aspects of my research. I was simultaneously impressed and humbled
by their expertise.
Other people to whom I would like to express my gratitude for their comments and
feedback include Hans Gruber (Germany), Isaac Linder (who wrote back in pen and ink
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from Russia), Michael Schlosser (University of Vienna, Austria), John Troyer
(University of Connecticut, USA), Matej Guid (FIDE Master, University of Ljubljana,
Slovenia), Muhidin Mulalic (International University of Sarajevo, Bosnia and
Herzegovina), A. C. Sukla (Sambalpur University, India) and the many unnamed
computer programmers, mathematicians and statisticians I have consulted with (and
learned from) over the Internet.
Special thanks to my colleagues, Uwe Dippel and Manjit Singh. The former for being
my (unofficial) academic mentor for several years and for his translation services
(German/French to English), and the latter for having shared with me many of his
experiences as a doctoral student. I would also like to thank the University of Malaya
staff (especially in the main library) for providing impeccable assistance and academic
resources. Even though they may never hear of it, my appreciation also goes to AT&T
Inc. for their ‘Natural Voices™’ technology, which enabled a computer to read this
entire thesis back to me in an almost human voice when it would have perhaps been too
much to ask of any human.
Very special thanks to Jaap van den Herik in the Netherlands for proofreading the final
draft of this thesis, and for his insightful comments. I wish you all the best, sir, on your
move from Universiteit Maastricht to Tilburg University, and sincerely appreciate the
time and effort you have spent on my behalf. Finally, I would like to thank my family
for their support and encouragement.
This research is supported by the University Tenaga Nasional research grant J510050123.
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For Gamers of the Future
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CONTENTS
ABSTRACT ..................................................................................................................... ii ACKNOWLEDGEMENTS ........................................................................................... iv LIST of FIGURES ......................................................................................................... xi LIST of TABLES ......................................................................................................... xiii ABBREVIATIONS ...................................................................................................... xiv CHAPTER 1: INTRODUCTION .................................................................................. 1
1.0 Preliminary ...................................................................................................... 1 1.1 Motivation ........................................................................................................ 3 1.2 Thesis Objectives ............................................................................................. 5 1.3 Thesis Scope ..................................................................................................... 6 1.4 Main Contributions of this Work .................................................................. 8 1.5 Thesis Organization ........................................................................................ 9 1.6 Summary of Research Questions ................................................................. 11
CHAPTER 2: LITERATURE REVIEW .................................................................... 13
2.0 Computational Research into Chess Aesthetics ......................................... 13 2.1 Emanuel Lasker and Aesthetics ................................................................... 15 2.2 Automatic Judging of Compositions ........................................................... 17 2.3 Principles of Beauty ...................................................................................... 21 2.4 Computer Chess Problem Composition ...................................................... 24 2.5 Elements of Beauty Classified ...................................................................... 27 2.6 Beauty Heuristics in a Game Engine ........................................................... 30 2.7 Computational Improvement of Chess Problems ...................................... 33 2.8 A Look at Methodologies Used in Other Domains ..................................... 38 2.9 Chapter Summary ......................................................................................... 42
CHAPTER 3: METHODOLOGY – Aesthetics in the Game ................................... 45
3.0 Components of the Research ........................................................................ 45 3.1 The Proposed Model of Aesthetics............................................................... 45 3.2 A Conceptual Framework for Aesthetics in the Game .............................. 46 3.3 An Examination of Aesthetics ...................................................................... 49
3.3.1 Composition Conventions ........................................................................... 50 3.3.2 Brilliancy in Real Games ............................................................................ 52 3.3.3 Principles of Aesthetics ............................................................................... 54
3.4 A Selection of Aesthetic Principles and Themes......................................... 58 3.5 A Formula for Cumulative Aesthetic Assessment ..................................... 61
3.5.1 The Development of Standard Evaluation Functions ................................. 62 3.5.2 Metrics and Properties Used in the Aesthetic Assessment ......................... 67
3.5.2(a) Piece Value and Piece Count .......................................................... 69 3.5.2(b) Distance, Piece Power, Mobility and Piece Field ........................... 71 3.5.2(c) Summary of Metrics and Properties ................................................ 73
3.5.3 A Note on Benchmarks ............................................................................... 74 3.6 A General Methodology for Developing Aesthetics Formalizations ........ 74 3.7 The Scope of Analysis Explained ................................................................. 78 3.8 Points of Evaluation (POE) .......................................................................... 80
3.8.1 The Moving Piece ....................................................................................... 81 3.9 Chapter Summary ......................................................................................... 82
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CHAPTER 4: METHODOLOGY – Aesthetic Principle Formalizations ................ 83 4.0 Formalizing the Seven Aesthetic Principles ................................................ 83 4.1 Violate Heuristics Successfully .................................................................... 85
4.1.1 Keep Your King Safe .................................................................................. 86 4.1.2 Capture Enemy Material ............................................................................. 87 4.1.3 Do Not Leave Your Own Pieces ‘En prise’ ................................................ 89 4.1.4 Increase Mobility of Your Pieces ................................................................ 90
4.2 Use the Weakest Piece Possible .................................................................... 91 4.3 Use All of the Piece’s Power ......................................................................... 92 4.4 Win with less Material .................................................................................. 93 4.5 Checkmate Economically ............................................................................. 94
4.5.1 Explanation of the Concept ......................................................................... 95 4.5.2 Features of Economy................................................................................... 96 4.5.3 The Economy Evaluation Function ............................................................. 97 4.5.4 The Process of Evaluation ......................................................................... 100 4.5.5 Validation .................................................................................................. 102
4.5.5(a) Compositions vs. Tournament Games .......................................... 102 4.5.5(b) Compositions vs. Tournament Games (Improved) ....................... 104 4.5.5(c) Testing against Human Assessment .............................................. 105
4.5.6 Minor Economical Differences ................................................................. 106 4.5.7 Paradoxical Economy................................................................................ 108 4.5.8 Perfect Economy ....................................................................................... 109
4.6 Sacrifice Material ........................................................................................ 110 4.7 Spread Out the Pieces (Sparsity) ............................................................... 112
4.7.1 Explanation of the Concept ....................................................................... 112 4.7.2 A Look at Possible Approaches ................................................................ 115 4.7.3 The Sparsity Evaluation Function ............................................................. 116 4.7.4 Validation .................................................................................................. 119
4.7.4(a) Sparsity and Piece Count .............................................................. 120 4.7.4(b) Sparsity and Piece Count (Alternative Method) ........................... 121 4.7.4(c) Sparsity and Piece Configuration .................................................. 122
4.7.5 Discussion ................................................................................................. 124 4.8 Points of Evaluation for the Aesthetic Principles ..................................... 125 4.9 Chapter Summary ....................................................................................... 125
CHAPTER 5: METHODOLOGY – Theme Formalizations .................................. 127
5.0 Formalizing the Ten Themes ..................................................................... 127 5.1 Fork .............................................................................................................. 128 5.2 Pin ................................................................................................................. 132 5.3 Skewer .......................................................................................................... 137 5.4 X-Ray ............................................................................................................ 139 5.5 Discovered/Double Attack .......................................................................... 142 5.6 Zugzwang ..................................................................................................... 146 5.7 Smothered Mate .......................................................................................... 149 5.8 Cross-check .................................................................................................. 150 5.9 Promotion .................................................................................................... 152 5.10 Switchback ................................................................................................... 154 5.11 Points of Evaluation for the Themes ......................................................... 155 5.12 Chapter Summary ....................................................................................... 156
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CHAPTER 6: EXPERIMENTAL RESULTS and DISCUSSIONS ....................... 158 6.0 The Six Experiments Performed ............................................................... 158 6.1 Experiment 1: Frequencies......................................................................... 160
6.1.1 Frequencies of the Themes........................................................................ 161 6.1.2 Frequencies of the Aesthetic Principles .................................................... 163 6.1.3 Discussion ................................................................................................. 164
6.2 Experiment 2: Evaluation of the Aesthetic Principles ............................. 165 6.3 Experiment 3: Evaluation of the Themes ................................................. 167 6.4 Experiment 4: Cumulative Evaluation ..................................................... 173
6.4.1 Aesthetic Principles Only .......................................................................... 178 6.4.2 Themes Only ............................................................................................. 180 6.4.3 Discussion ................................................................................................. 182
6.5 Experiment 5: Conformity to Authoritative Human Assessment .......... 183 6.6 Experiment 6: Correlation with Human Assessment .............................. 186
6.6.1 Survey 1 (Mixed) ...................................................................................... 190 6.6.2 Survey 2 (Mixed, Discrete Evaluations) ................................................... 192
6.6.2(a) Levels of Agreement ..................................................................... 194 6.6.3 Survey 3 (Tournament Games) ................................................................. 195 6.6.4 Survey 4 (Compositions) .......................................................................... 197 6.6.5 Survey Conclusions ................................................................................... 198
6.7 Chapter Summary ....................................................................................... 202 CHAPTER 7: CONCLUSION ................................................................................... 205
7.0 Preliminary .................................................................................................. 205 7.1 Thesis Summary .......................................................................................... 205 7.2 Thesis Contributions ................................................................................... 208 7.3 Implications of the Research ...................................................................... 209 7.4 Directions for Further Work...................................................................... 212
REFERENCES ............................................................................................................ 217 APPENDIX A: CHESS RULES ................................................................................ 233
1.0 Introduction to the Game ........................................................................... 233 1.1 Movement of the Pieces .............................................................................. 234
1.1.1 Rook .......................................................................................................... 235 1.1.2 Bishop ....................................................................................................... 235 1.1.3 Queen ........................................................................................................ 236 1.1.4 Knight ........................................................................................................ 236 1.1.5 King ........................................................................................................... 237
1.1.5(a) Castling ......................................................................................... 237 1.1.6 Pawn .......................................................................................................... 239
1.2 Check, Checkmate and Stalemate ............................................................. 240 1.3 Other Rules .................................................................................................. 242
1.3.1 Resignation and Draws ............................................................................. 242 1.3.2 Repetition of Positions .............................................................................. 243 1.3.3 50-Move Rule ........................................................................................... 243 1.3.4 Touching Pieces ........................................................................................ 244
1.4 Chess Notation ............................................................................................. 244 1.4.1 Board Notation .......................................................................................... 246
APPENDIX B: GLOSSARY of CHESS TERMS .................................................... 248
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APPENDIX C: EXAMPLE POSITIONS ................................................................. 253 Subsection 3.3.1 ....................................................................................................... 253 Subsection 3.3.3 ....................................................................................................... 254 Subsection 3.5.1 ....................................................................................................... 255 Section 4.2 ................................................................................................................ 256 Subsection 4.5.4 ....................................................................................................... 257 Section 5.1 ................................................................................................................ 257 Section 5.2 ................................................................................................................ 259 Section 5.4 ................................................................................................................ 260 Section 5.5 ................................................................................................................ 261 D (Refer Appendix) ................................................................................................. 263
APPENDIX D: CHESTHETICA .............................................................................. 264 APPENDIX E: PSEUDOCODE ................................................................................ 271
Subsection 4.1.2 ....................................................................................................... 271 Subsection 4.1.3 ....................................................................................................... 272 Section 4.3 ................................................................................................................ 272 Section 4.5 ................................................................................................................ 273 Section 4.7 ................................................................................................................ 274 Section 5.1 ................................................................................................................ 274 Section 5.2 ................................................................................................................ 276 Section 5.4 ................................................................................................................ 278 Section 5.5 ................................................................................................................ 279 Section 5.6 ................................................................................................................ 281
APPENDIX F: SURVEY DATA ............................................................................... 282
1.0 Overview of the Surveys ............................................................................. 282 1.1 Instruction Set (Surveys 1, 3 and 4) ........................................................... 283
1.1.1 Instruction Set (Survey 2) ......................................................................... 285 1.2 The Combinations Rated ............................................................................ 288 1.3 The Combinations Rated (PGN Compatible) ........................................... 300 1.4 Control Questions ....................................................................................... 305
1.4.1 Survey 1 .................................................................................................... 307 1.4.2 Survey 2 .................................................................................................... 308 1.4.3 Survey 3 .................................................................................................... 309 1.4.4 Survey 4 .................................................................................................... 310
1.5 Respondent Ratings .................................................................................... 311 1.6 Screen Captures .......................................................................................... 319
SELECTED PUBLICATIONS .................................................................................. 321
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LIST of FIGURES
Figure 3.1 Concept of Aesthetics in Chess ............................................................... 48 Figure 3.2 Layers of an Aesthetic Evaluation Function ........................................... 67 Figure 4.1 Scores for Violation of ‘Keep Your King Safe’ ..................................... 86 Figure 4.2 Maximum ‘Control Fields’ for the Chessmen ........................................ 98 Figure 4.3 Economy Scores of Checkmate Positions ............................................. 102 Figure 4.4 Economy Scores for Compositions and Tournament Games ............... 103 Figure 4.5 Economy Scores for ‘Improved’ Positions ........................................... 105 Figure 4.6 Minor Economic Improvements to a Position ...................................... 107 Figure 4.7 Economy Paradox ................................................................................. 108 Figure 4.8 Highly Economical Checkmates ........................................................... 110 Figure 4.9 Sparsity in Chess Compositions ............................................................ 113 Figure 4.10 Sufficient Sparsity (Constructed Positions) .......................................... 114 Figure 4.11 Sparsity Scores of Chess Positions from Tournament Games .............. 118 Figure 4.12 Sparsity Scores of Go Positions ............................................................ 119 Figure 4.13 Sparsity Values of 1,000 Random Game Positions .............................. 120 Figure 4.14 Sparsity Values of 1,000 Random Game Positions (Alternate) ............ 122 Figure 5.1 The Fork ................................................................................................ 129 Figure 5.2 The Pin .................................................................................................. 133 Figure 5.3 Aesthetic Assessment of the Pin ........................................................... 136 Figure 5.4 The Skewer............................................................................................ 137 Figure 5.5 Aesthetic Assessment of the Skewer..................................................... 138 Figure 5.6 The X-ray .............................................................................................. 139 Figure 5.7 Aesthetic Assessment of the X-Ray ...................................................... 142 Figure 5.8 The Discovered/Double Attack ............................................................. 143 Figure 5.9 Aesthetic Assessment of the Discovered Attack ................................... 144 Figure 5.10 The Zugzwang ....................................................................................... 148 Figure 5.11 The Smothered Mate ............................................................................. 149 Figure 5.12 Aesthetic Assessment of the Cross-check ............................................. 151 Figure 5.13 The Saavedra Position ........................................................................... 153 Figure 6.1 Frequencies of Themes in the Combinations ........................................ 161 Figure 6.2 Frequencies of Aesthetic Principles in the Combinations..................... 163 Figure 6.3 Cumulative Aesthetic Scores for Combinations ................................... 174 Figure 6.4 Highest Scoring Combinations (a) COMP, (b) TG ............................... 175 Figure 6.5 Lowest Scoring Combinations (a) COMP, (b) TG ............................... 177 Figure 6.6 Cumulative Scores Based on Aesthetic Principles Only ...................... 179 Figure 6.7 Cumulative Scores Based on Themes Only .......................................... 181 Figure A.1 The Initial Position of the Pieces .......................................................... 234 Figure A.2 Movement of the Rook.......................................................................... 235 Figure A.3 Movement of the Bishop ....................................................................... 235 Figure A.4 Movement of the Queen ........................................................................ 236 Figure A.5 Movement of the Knight ....................................................................... 236 Figure A.6 Movement of the King .......................................................................... 237 Figure A.7 Before and after Castling ...................................................................... 238 Figure A.8 Castling Illegal for Both White and Black ............................................ 238
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Figure A.9 Movement of the Pawn ......................................................................... 239 Figure A.10 En passant ............................................................................................. 240 Figure A.11 Check..................................................................................................... 241 Figure A.12 Checkmate and Stalemate ..................................................................... 242 Figure A.13 The Chessboard and its Coordinates ..................................................... 244 Figure C.1 A Typical ‘Logical’ School Composition ............................................. 253 Figure C.2 J. Mintz, The Problemist, 1982, Helpmate in 3 (Black to Play) ........... 254 Figure C.3 The ‘Immortal Game’ (after 17. … Qxb2)............................................ 254 Figure C.4 Kasparov vs. Deep Junior, Game 5, New York, 2003 .......................... 255 Figure C.5 Deep Blue vs. Kasparov, Game 6, New York, 1997............................. 255 Figure C.6 Two-way Discovered Checkmate ......................................................... 256 Figure C.7 Two-Phase Piece Removal (Economy) ................................................. 257 Figure C.8 Activated Fork ....................................................................................... 257 Figure C.9 Repeated Fork ....................................................................................... 258 Figure C.10 Immobilizing the Queen with a Two-way Pin ...................................... 259 Figure C.11 A Three-way Pin (Bxd5) ....................................................................... 259 Figure C.12 Negative Evaluation for the Pin after 1. Qg2 ........................................ 260 Figure C.13 A Double X-ray with 1. Bxd4 ............................................................... 260 Figure C.14 Castling as a Discovered Attack Manoeuvre (0-0#) ............................. 261 Figure C.15 Double-Discovered Attack (after 1. Ndf4+) ......................................... 261 Figure C.16 Stalemate in 3 ........................................................................................ 263 Figure D.1 The Main Interface to CHESTHETICA ............................................... 265 Figure D.2 The ‘About Box’ ................................................................................... 266 Figure D.3 The Aesthetics Evaluation Panel ........................................................... 267 Figure D.4 The Thematic Frequency Chart ............................................................. 268 Figure D.5 Aesthetic Principle and Theme Selection ............................................. 268 Figure D.6 The Mate Solver .................................................................................... 269 Figure F.1 Survey 1: Control Question 1 ................................................................ 307 Figure F.2 Survey 1: Control Question 2 ................................................................ 307 Figure F.3 Survey 2: Control Question 1 ................................................................ 308 Figure F.4 Survey 2: Control Question 2 ................................................................ 308 Figure F.5 Survey 3: Control Question 1 ................................................................ 309 Figure F.6 Survey 3: Control Question 2 ................................................................ 309 Figure F.7 Survey 4: Control Question 1 ................................................................ 310 Figure F.8 Survey 4: Control Question 2 ................................................................ 310 Figure F.9 Survey 3: Screen Capture 1 ................................................................... 320 Figure F.10 Survey 3: Screen Capture 2 ................................................................... 320
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LIST of TABLES
Table 3.1 General Aesthetic Principles, Conventions and Brilliancy Compared .... 55 Table 3.2 Refined Aesthetic Principles and Themes ............................................... 60 Table 3.3 Metrics and Properties Used .................................................................... 73 Table 3.4 Points of Evaluation in a Combination .................................................... 80 Table 4.1 Points of Evaluation for the Aesthetic Principles .................................. 125 Table 5.1 X-ray Defensive Capabilities ................................................................ 141 Table 5.2 Points of Evaluation for the Themes ..................................................... 156 Table 6.1 Average Scores for Aesthetic Principles ............................................... 165 Table 6.2 Standard Deviations of Average Aesthetic Principle Scores ................ 166 Table 6.3 Significance of Diff. between Mean Aesthetic Principle Scores .......... 167 Table 6.4 Average Scores for the Themes ............................................................. 168 Table 6.5 Standard Deviations of Average Scores for Themes ............................. 169 Table 6.6 Significance of Mean Differences of Theme Scores ............................. 170 Table 6.7 Average Cumulative Aesthetic Scores for the Combinations ............... 173 Table 6.8 Average Cumulative Aesthetic Scores for Aesthetic Principles Only .. 178 Table 6.9 Average Cumulative Aesthetic Scores for Themes Only ...................... 180 Table 6.10 Human vs. Computer Assessment (COMP+TG Combinations) .......... 190 Table 6.11 Human Assessment vs. Computer (Discrete Evaluations) .................... 193 Table 6.12 Level of Human Agreement with Computer Assessment ..................... 194 Table 6.13 Human vs. Computer Assessment (TG Only)....................................... 196 Table 6.14 Human vs. Computer Assessment (COMP Only)................................. 198 Table 6.15 Summary of Human-Computer Assessment Correlations .................... 199 Table 6.16 Summary of Positive Correlations ........................................................ 201 Table A.1 The Chessmen ....................................................................................... 233 Table A.2 Shorthand Notation ................................................................................ 246
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ABBREVIATIONS
1T One-tailed
2T Two-tailed
AI Artificial Intelligence
APP Use all of the Piece’s Power
COMP Chess Composition Combinations
DDA Discovered/Double Attack
FIDE Fédération Internationale des Échecs
FEN Forsyth-Edwards Notation
FM FIDE Master
GM Grandmaster
ICP Improver of Chess Problems, The
IM International Master
LOA Level of (Human) Agreement
LOD Level of (Computational) Distinction (between Scores)
PGN Portable Game Notation
SD Standard Deviation
SL Significance Level (of)
TG Tournament Game Combinations
TTUV Two-sample t-test assuming Unequal Variances
VH Violate Heuristics (Successfully)
WPP Use the Weakest Piece Possible
WWLM Win with Less Material
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CHAPTER 1: INTRODUCTION
1.0 Preliminary
Research into games is important, especially in the field of AI. The following paragraph
articulates the main reasons quite well.
“There are two principal reasons to continue to do research on games. . .
First, human fascination with game playing is longstanding and pervasive.
Anthropologists have catalogued popular games in almost every culture. . .
Games intrigue us because they address important cognitive functions. . . The
second reason to continue game-playing research is that some difficult games
remain to be won, games that people play very well but computers do not.
These games clarify what our current approach lacks. They set challenges for
us to meet, and they promise ample rewards.” (Epstein, 1999)
A zero-sum perfect information game (sometimes with a hyphen between ‘perfect’ and
‘information’) is one in which a player gains at the equal expense of others and where
every player knows the results of all the previous moves. Examples include noughts and
crosses, checkers, chess and Go (in scientific literature, there is some common
agreement that the game be referred to using a capital letter to differentiate it from the
English verb ‘go’). These are games where it is theoretically possible to build a
computational move ‘tree’ of all the positions that could occur and thus facilitate perfect
play. However, for games such as Go and chess the number of possible positions to
examine is too large even for computers. It is estimated that there are approximately
1046 legal positions in chess and around 10170 in Go (Chinchalkar, 1996; Tromp and
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Farnebäck, 2006). Even so, good evaluation functions and efficient search techniques
allow computers to play such games, with the current and notable exception of Go, quite
well (Levy and Newborn, 1992; Campbell et al., 2002; Walczak, 2003; Hauptman and
Sipper, 2007; Hsu, 2007).
One of the most popular research domains in this respect is chess. The beginnings of
chess are obscure but it is thought to have originated in northern India around 600 AD
and spread mainly through traders to other parts of the world (Eales, 2002; Shenk,
2006). The most widely played version today is Western or international chess and is
regulated by the Fédération Internationale des Échecs (FIDE) or World Chess
Federation. The rules of international chess have remained essentially the same since
1475 (Hooper and Whyld, 1996). Ever since Claude Shannon (1950) wrote his seminal
paper on how a computer could be programmed to play chess; researchers,
programmers and especially the public have been fascinated at the prospect of
computers playing the game at the same level of human experts or better (Grier, 2006).
The approach suggested is often also credited to Alan Turing (1953). The hope was that
whatever methods achieved this might shed some light onto the mechanics of human
thought processes because one must be thinking in order to play the game (Newborn,
1997). Computer chess programs today play at the grandmaster (GM) level and have
even beaten the world champion by relying mainly on brute-force (i.e. exhaustive or
nearly exhaustive searching of relevant parts of the game tree) which is different from
how humans play (Hsu, 2004). Even though this is not exactly what AI researchers were
hoping for (Hendler, 2006), research into the game has provided benefits and insights
into other areas (see section 2.0).
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However, there is an important aspect of chess that computers are quite poor at. This is
aesthetics and it is one of the main reasons humans play (Kasparov, 1987; Damsky,
2002; Dossi, 2005). Computers cannot tell a beautiful move combination or game from
a regular or unattractive one the way humans can. Chess was chosen as a suitable
domain of research because its aesthetic aspect (especially prominent in chess
compositions) is well established in the literature, more so than Go or any other zero-
sum perfect information game.
Real games (e.g. in tournaments) are also known to exhibit aesthetics, and ‘brilliancy’
prizes are sometimes awarded to such games. With chess programs playing on a level
equal to - and in some cases greater than - the best human players, the time seems ripe
for focusing on the aesthetics of the game. Since humans strive for, and appreciate
beauty in chess, it would be valuable if computers could recognize it in a way that is
comparable to humans given that machines are able to analyze many more positions in
the game tree than humans.
1.1 Motivation
A zero-sum perfect information game is a good place where complex ideas can be
experimented with because in theory, such games are finite and particularly amenable to
computation. This is the reason chess has for many decades been the subject of much
research in various fields (see section 2.0). However, the main focus has always been on
how to make computers play the game on a level equal to, or exceeding that of the best
human players. This has, since the late 1990s, been demonstrated (see subsection 3.5.1).
Even commercially available computer programs today play at the grandmaster level.
Researchers have therefore, in this respect, now generally shifted to more complex
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games like Go (Wu and Baldi, 2007). The following six reasons served as motivation
for this research.
• Aesthetics in chess has received little attention in AI despite it being an
important part of the game that matters to players. Humans, unlike
computers, do not play solely to win. They also want their games and
compositions to be beautiful and fascinating. A computational model for
aesthetics in the game would therefore benefit humans and enhance the
capabilities of existing chess programs.
• Research into automatic chess problem composition has not accounted for
aesthetics in a meaningful way or perhaps at all (see sections 2.2 and 2.4). It
neither separates aesthetics from composition convention (see Appendix B)
nor takes into account much of the knowledge that is available in chess
literature on the subject of aesthetics.
• Existing formalizations (usually in the form of an evaluation function
represented using a mathematical formula) on chess ‘quality’ typically use
fixed values attributed to aesthetic principles and chess themes. These do not
reflect the variations possible in such principles and themes (e.g. a different
piece configuration of the same theme) in a way that is both flexible yet
consistent. As a result, aesthetics is not accounted for reasonably in
compositions (see section 2.7), and even less in actual games.
• Other similarly complex zero-sum perfect information games such as Go and
up to a thousand other chess variants also have an aesthetic dimension that
has not benefited from computational analysis (Pritchard, 2000b). An
aesthetics model for chess could, in principle, be extended or adapted to
these games as well for the benefit of humans. Chess variants for example,
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usually vary in only one respect such as the board size or additional piece
types.
• Aesthetic models have been developed with some success in less discrete
domains such as art (e.g. photographic images, paintings) and music
(Machado and Cardoso, 1998; Golub, 2000; Manaris et al., 2002a, 2002b;
Datta et al., 2006). A more reliable model could perhaps be developed for a
theoretically finite domain like chess. This in turn could inspire the
development of better models in those domains and others since chess is
often used or referenced in many areas of research (see section 2.0).
• A personal interest in the game of chess and over 20 years of active playing
experience, combined with an equal interest and level of experience with
computer programming. The prospect of developing a computer program –
probably the first of its kind - capable of recognizing beauty in chess based
on a viable model was therefore also a motivator.
Due to all the reasons mentioned above, a discrete computational aesthetics model for a
zero-sum perfect information game like chess was deemed worthy of investigation. The
term ‘discrete’ signifies the distinct yet synergetic components of the proposed model
(see section 3.1).
1.2 Thesis Objectives
The objectives of this thesis are as follows.
1. To study chess (as a zero-sum perfect information game) and its relevant
literature on aesthetics to identify the pertinent issues.
6
2. To propose a model that makes aesthetic evaluation in the game
computationally feasible.
3. To derive formalizations for a selection of aesthetic principles.
4. To derive formalizations for a selection of chess themes.
5. To develop a computer program incorporating those formalizations for the
purpose of performing relevant experiments.
6. To test the viability of the model through experimentation in terms of
aesthetic recognition in the game and positive correlation with human
aesthetic assessment.
1.3 Thesis Scope
The scope of this thesis is as follows.
1. Review the relevant literature on chess with emphasis on its aesthetic aspect.
2. Review and evaluate existing methodologies that have attempted to address
aesthetics in the game computationally. Include also research that did not
actually address aesthetics where it would have been pertinent. In the interest
of a general context, briefly review methodologies of aesthetic evaluation in
other domains such as art and music.
3. Propose a conceptual framework for aesthetics in the game which is a way of
thinking about it that can guide proper investigation. Contrast with the
current practice of conflating composition convention with aesthetics.
4. Investigate composition conventions and brilliancy in real games. Identify
the areas in which they overlap with established aesthetic principles, as also
described in chess literature.
7
5. Propose a selection of aesthetic principles and themes for evaluation. These
include those that apply, as far as possible, equally to both domains.
6. Propose a formula for cumulative aesthetic assessment in a move
combination. Explain the choice of metrics and properties to be used.
7. Present a general methodology for developing formalizations for aesthetic
principles and themes in the game.
8. Define and explain the chosen scope of mate-in-3 combinations and the
points of evaluation for each selected aesthetic principle and theme. This is
to facilitate experimentation.
9. Propose formalizations for the selected aesthetic principles and themes in the
game with explanations about the logic behind their design. Include
discussions about their strengths and limitations.
10. Develop a computer program that incorporates these formalizations to make
experimentation feasible. It should possess features which make a variety of
experiments involving compositions and real games possible.
11. Validate the aesthetics model through experimentation in terms of its ability
to recognize aesthetics in the game computationally.
12. Demonstrate that the computational assessment of aesthetics in the game,
within the chosen scope of mate-in-3 combinations, correlates positively
with human chess player aesthetic assessment.
This thesis explores aesthetic principles that pertain to the game of international chess in
general but limits experimental studies to mate-in-3 combinations. Aesthetic principles
that apply mainly to other domains such as art and music are beyond the scope of this
thesis.
8
1.4 Main Contributions of this Work
The key contributions of this research include the following.
1. To review and examine aesthetic principles (and themes) in the zero-sum
perfect information game of international chess as described in its relevant
literature.
2. To propose a conceptual framework for aesthetics as a way of thinking about
aesthetics in the game. This framework isolates aesthetics as a component
not exclusive to compositions or real games. It makes aesthetics more
computationally amenable and easier to apply to both domains in a way that
does not conflate with composition convention or brilliancy (in real games).
3. To propose a formula for cumulative aesthetic assessment in a move
sequence or combination. It is based on the idea that dynamic formalizations
for aesthetic principles and themes, in summation, can represent the aesthetic
content of a combination. These are in turn based on metrics and properties
inherent to the game. This permits the use of aesthetic evaluation in addition
to other forms of computational assessment in the game such as composition
convention and standard game-playing heuristics.
4. To present a general methodology for developing aesthetics formalizations
in the game. Using this approach (and the concept of benchmarks),
formalizations for other aesthetic principles and themes can be developed in
a similar way to those developed for this research.
5. To propose a set of dynamic formalizations (and explain the logic behind
their individual designs) for a selection of aesthetic principles and themes.
9
These represent the common ground of aesthetics between the domains of
chess compositions and real games.
6. To devise a few novel experiments for validating aesthetic recognition in the
game based on the proposed computational aesthetics model.
7. To develop a computer program incorporating the aesthetics model (and all
the formalizations) for experimental purposes. Manual evaluation is
complicated and prone to error, especially for bulk analysis. This program
can be used by other researchers to save time in evaluating aesthetics based
on the model.
8. To evaluate the proposed aesthetics model in terms of its ability, when
implemented, to recognize aesthetics in the game.
9. To test the computational aesthetic recognition capability of the model for
positive correlation with human chess player assessment.
The results of a variety of experiments suggest that the aesthetics model can be used to
recognize aesthetics in the game of chess within (at least) the scope of mate-in-3
combinations. The aesthetic scores produced by a computer program based on the
model also correlate well with human chess player aesthetic assessment.
1.5 Thesis Organization
The detailed structure of this thesis is as follows. Chapter 2 reviews the relevant chess
and scientific literature pertaining to aesthetics in the game. It generally illustrates the
importance of aesthetics in chess over the last century, and the attempts that have been
made to bring that concept into the computational domain. The chapter also briefly
10
reviews methodologies applied to gauge aesthetics in other domains such as art and
music.
Chapter 3 introduces the proposed model of aesthetics. The components of the model
include an examination of aesthetics from the perspective of problem composers and
players, as described in the relevant chess literature. A common ground of aesthetics is
identified as the focus of this research. A selection is made of aesthetic principles and
themes that fall within that common ground. The chapter also presents a formula for
cumulative aesthetic assessment and an explanation of the metrics and properties used.
The methodologies behind standard evaluation functions in the game are reviewed
before one is proposed for developing aesthetic evaluation functions (i.e. related to the
selected principles and themes). The scope of analysis (for experimental purposes) is
then explained, followed by a description of the points of evaluation in a combination.
Chapter 4 details the actual evaluation functions for the seven selected aesthetic
principles. A detailed description of each principle and the logic behind the design of its
function are presented. Diagrams, examples, and experimental validation are provided,
where appropriate. Chapter 5 details the formalizations for the ten selected themes,
similar to the previous chapter. Chapter 6 presents and explains all the experiments
performed to validate the proposed model. Included are analytical discussions of the
results. Chapter 7 concludes with a summary of the thesis, its contributions, a section on
the research implications, and directions for further work in the area.
Appendix A explains the rules of international chess and how to read its algebraic
notation. Appendix B is a glossary of chess-related terms found in this thesis. Appendix
C contains diagrams of example positions referenced primarily in the main text.
11
Appendix D features specifications and information about the computer program (i.e.
CHESTHETICA) developed for this research. Appendix E shows the essential
pseudocode for implementing many of the proposed aesthetic formalizations. Appendix
F contains the survey questionnaires used and the relevant raw data that was collected.
Unless inclusive of the word, ‘Appendix’, references to specific parts of this document
are given according to chapter, section or subsection; e.g. chapter 3, section 3.5,
subsection 3.5.2 and subsection 3.5.2(d). General formulas (in the form of equations)
are numbered sequentially according to section but instantiations of those equations
(such as sample calculations) are not numbered. Chess move notation and board
coordinates in line with other text in the main document are given in bold (e.g. a5).
Example positions are sometimes given in FEN (Forsyth-Edwards Notation) within the
main text (with a reference to a corresponding diagram in Appendix C). The words
‘thesis’ and ‘research’ are sometimes used interchangeably to refer to the work
presented in this document.
1.6 Summary of Research Questions
In principle, this thesis attempts to answer two research questions.
1) Can aesthetics in chess (within a specific scope) be recognized
computationally?
2) If so, do the computational evaluations correlate positively with human chess
player aesthetic assessment?
12
Six experiments were performed to answer both these questions (see chapter 6) with
promising results. Question 1 is confirmed to the extent of mate-in-3 combinations (a
reasonable scope of analysis in the game) and question 2 to the extent involving
competent (not necessarily expert) human chess players; consistent with what is
necessary for aesthetic appreciation in the game.
13
CHAPTER 2: LITERATURE REVIEW
2.0 Computational Research into Chess Aesthetics
Computational research into the aesthetics of chess is relatively scarce. This may be due
to the emphasis over the last few decades on getting computers to play the game ever
more proficiently (Adelson-Velskiy et al., 1970; Hartmann, 1987a, 1987b; Heinz, 1997;
Walczak, 2003). It may also be due to the assumption that there is no reliable way of
quantifying beauty in the game (le Grand, 1986). In the first case, computers have today
advanced to the level of world-class players and are even used by most of them for
training purposes (Muller, 2002; Ross, 2006; Sukhin, 2007); this likely includes the use
of game databases and not just chess-playing programs (Campitelli and Gobet, 2007).
Therefore, rather than just moving to more complex games like Go (Hsu, 2007;
McCarthy, 2007), more emphasis can now be placed on aesthetics in chess. In the
second case, there exists substantial material on the subject in chess literature (see the
following sections) to base a computational model on. Since human players and
problem composers value beauty in the game, the idea of computational recognition of
beauty is worthy of investigation. There is also room for its application in existing
research which, currently address chess aesthetics superficially (see sections 2.6 and
2.7). Computational approaches to art forms are not unfeasible and are likely to become
more common in the future (Boden, 2007).
It is said that there are more books written on chess than books on all other games
combined (Jonsson, 2006). With over 700 million players worldwide (Polgar, 2005), it
is arguably the most popular game in the world (Wolff, 2001). It is also recognized as a
14
sport by the International Olympic Committee (IOC, 2008). Benjamin Franklin wrote
about the benefits of chess as far back as 1779 in his article, ‘On the Morals of Chess’
(Shenk, 2006). Investigations into chess have had many applications within and outside
of AI including molecular computing (Cukras et al., 1999; Faulhammer et al., 2000),
automated theorem proving (Newborn, 2000), computer music composition (Friedel,
2006), machine reading (Etzioni et al., 2007), cognitive development (O’Neil and Perez,
2007), treatment of psychiatric illness (Cavezian et al., 2008) and children education
(Ferreira and Palhares, 2008; AF4C, 2008). Such applications are not always predictable
so it is conceivable that this research could also be of interest or reference to researchers
in other fields or related ones.
This chapter reviews the more important and relevant contributions to computational
aesthetics in chess and scientific literature over the last century. It is a relatively recent
account, given for example, that documents featuring chess compositions date back over
1,000 years (Al-Adli, 9th century). Whole books have been written on chess since the
15th century (Axon, 1474). A review spanning the last 85 years or so is necessary to
illustrate how aesthetic principles have been described by experts and researchers prior
to and since the computer age. The reviews are arranged in chronological order for
proper perspective with a summary in section 2.9. A glossary of chess terms is provided
in Appendix B for reference. Even though it may not be directly applicable, in the
interest of a general context, section 2.8 presents a brief discussion about methodologies
relating to aesthetics used in other domains.
15
2.1 Emanuel Lasker and Aesthetics
Former world chess champion and mathematician Emanuel Lasker was one of the first
to write about aesthetics in chess explicitly. He maintained the world title for 27 years
starting in 1894, the longest ever held by a world champion. In his book, “Lasker’s
Manual of Chess” – originally written in 1925 - he devoted a chapter to the subject
entitled, ‘The Aesthetic Effect in Chess’ and stressed on the concept of ‘achievement’
and ‘correctness’ (Lasker, 1960). ‘Achievement’ means that beauty in the game had to
have some kind of positive result such as winning material, controlling more space on
the board, or checkmating the opponent, whereas ‘correctness’ implies that the method
of achievement be absolutely necessary and unequivocal. In other words, there had to be
no possible escape or defence by the opponent and no better way of attaining the same
achievement (e.g. in fewer moves). He also stated that in order to appreciate aesthetics
in chess, one need only to understand the game and not be a master himself.
Hence, the average player can just as easily derive pleasure from beautiful games and
compositions. He termed the pleasure spectators derived from the game – due to
witnessing moves they would call ‘brilliant’ or ‘beautiful’ - their ‘aesthetic valuation’.
This valuation was based on their immediate perception of a move’s brilliance. As a
result, some manoeuvres that were not, in fact, ‘correct’ (as would be revealed in the
post-mortem analysis of the game) elicited a high aesthetic valuation until their
incorrectness was discovered. In such cases, the valuation might diminish a little unless
it was suspected that the players had done so intentionally to fool the audience; in which
case it would diminish entirely. In tournament games, time constraints place a
considerable burden on players so all the variations of an attractive move combination
16
may not have been worked out properly (Harreveld et al., 2007). In compositions there
is no excuse.
So aesthetically, the occurrence of somewhat ‘incorrect’ brilliant moves in real games is
condoned whereas in compositions, they must withstand the most rigorous analysis.
Modern computer programs as a result have revealed weaknesses and flaws in many old
games and compositions that were once thought to be spectacular. Lasker showed many
examples of beautiful combinations from both tournament games and compositions.
Most of them were forced mates between 2 and 4 moves but others were longer and
more complex. One example in his book by an unknown composer featured a ‘forced
draw’ by White (despite having a significantly inferior army) that took 17 moves to
complete. In it, a pair of knights chased the enemy king around the board before the
initial position finally repeated itself. A game is considered drawn if the same position
recurs three times (see Appendix A, subsection 1.3.2).
Lasker was not specific about what he considered the tangible constituents of aesthetics
in chess and relied mainly on his experience with the game as opposed to
experimentation. His concepts of ‘achievement’ and ‘correctness’ are useful precepts
for a framework for aesthetics even though he proposed no formalizations himself. This
is understandable given the period in which his book was written, i.e. well before the
advent of computers. It is quite possible that even the idea of computing aesthetics in
the game was too controversial to be taken seriously. Lasker made no distinction
between the aesthetics of real games and compositions and thus did not take into
account composition conventions. This supports the idea that aesthetics transcends
either domain, at least in cases where the rules are the same (i.e. not including chess
17
variants). Even so, it possibly overlooks situations where some conventions could also
apply aesthetically to real games.
Lasker made the important observation that aesthetic appreciation is not an experience
limited to masters even though having an understanding of the game would imply a
certain level of competence as a player. While master players may be needed to identify
the principles of aesthetics, others can also appreciate them. Assessing computational
evaluation of aesthetics in terms of positive correlation with human perception would
therefore not necessitate the involvement of experts. In fact, it would probably be more
useful to exclude or not focus on them since they form only a small minority of the
chess community.
The psychological aspects of aesthetic perception Lasker suggested are interesting but
difficult to gauge computationally because they rely upon the intentions of players and
subjective valuation by spectators that potentially change based on those intentions.
These are neither computationally amenable nor within the scope of this thesis. In
summary, Lasker provides a good starting point on how to approach the question of
aesthetics in the game but his contribution lacks the building blocks (i.e. discrete
components) required for a computational model.
2.2 Automatic Judging of Compositions
Vaux Wilson - a chess composition judge and author - researched, proposed, and
refined over the course of 20 years, a method of evaluating the aesthetic and strategic
elements of chess problems through a scientific approach (Wilson, 1959, 1969, 1978).
The intention was to provide a logical basis for compositions to be judged in
18
tournaments because many composers felt that judges were too arbitrary or subjective
when choosing a winning composition. The method was supposedly similar to a much
earlier and obscure system proposed by Harley (1919) that was limited to two-move
problems, but no reference was made to that work.
Wilson identified exclusively nine basic ways or ‘strategic elements’ in which a moving
piece might influence the game. It could:
1. capture an opposing piece or sacrifice itself;
2. give check to the enemy king;
3. guard or abandon guarding a square;
4. move into a position where it could gain access to another square;
5. block or unblock a square;
6. castle;
7. move off, on or along a line;
8. open or close a line of check, or the guard of one square;
9. pin or unpin a piece.
These strategies were valued (typically between 2 and 10 points) based on the number
of pieces and squares involved. A ‘line’ was determined as consisting of 3 squares.
Since nothing else could possibly happen on the board when the pieces move in a chess
problem, the cumulative value of these strategies was considered to be inclusive of
aesthetics and any other impression one might obtain from a composition. Wilson’s
actual system incorporated several rules and exceptions that compensated for
composition conventions, e.g. with respect to ‘keys’ and ‘tries’ (see Appendix B). In
addition, a concept of economy was also evaluated and added for a final score.
19
Economy was calculated as the sum of strategic scores divided by the number of white
pieces. In general, the system was designed exclusively with compositions in mind, and
not real games, even though aesthetics is perceived in both (Humble, 1993, 1995;
Ravilious, 1994).
Wilson’s system was limited to chess problems but directly applicable to the many
different types such as orthodox mates, selfmates, helpmates, endgame studies, and
fairy problems (of any move length). He tested it on over 7,000 problems with
satisfactory results (i.e. compositions he perceived to be better scored higher) and the
system was used in a few composition tournaments. A significant positive correlation
with human judge evaluations was never demonstrated and probably because the system
was developed to address that very problem.
However, composers soon stopped using it mainly because human judges could not be
entirely replaced, as Wilson intended (le Grand, 1986). For example, compositions often
feature characteristic themes that could not properly be accounted for using the
strategies. At the time, Wilson was in the process of having a computer program
developed that incorporated his method and made calculations for composers even
easier. It is not known if this program was ever completed.
Wilson’s system conflated aesthetics with composition convention and this failed to
account for either one sufficiently, especially the former. His identification of chess
strategies was perhaps one of the earliest and most systematic approaches to the
problem of evaluating compositions (which nevertheless include an aesthetic
dimension) but suffered from oversimplification. There are two essential points that
illustrate this. First, the values attributed to each strategy were not adequately justified
20
and seemed to have stemmed from his experience as a composition tournament judge,
thereby limiting their applicability to little beyond compositions. These values also
failed to account for the variety of piece configurations possible within each strategy.
Second, metrics inherent to the game such as the number of pieces and distance (in
terms of squares on the board) were employed as the basis of some of the strategies but
were not incorporated in their universal form, e.g. defining a line as just 3 squares long
and not using the standard Shannon (1950) value of each piece (see subsection 3.5.2(a)).
These were apparently done to simplify calculations that, at the time, had to be
performed largely without the aid of computers.
Wilson tested his system experimentally on many contest-winning and generic problems
but his results were not very reliable because the intention was to replace the existing
paradigm of composition evaluation with a new, unbiased, and systematic one. What
seemed reasonable to him may not, in fact, have been so to the wider composition
community despite the best intentions to eliminate subjectivity. The scope of application
was also too large because it may be unreasonable to assume that human aesthetic
perception or appreciation of compositions remains constant despite the length of moves
or type of problem involved.
The rejection of his system by the composition community, however, is not necessarily
an indication of a failed approach. A more limited scope and flexible set of established
values (attributed to the strategies) would probably have improved it. In general, Wilson
introduced a scientific approach to aesthetics in the game even though not dealing with
it directly, and he showed that the evaluation of subjective aspects in chess is effective
to some degree using identified and quantifiable strategies or principles.
21
2.3 Principles of Beauty
The psychologist Stuart Margulies was perhaps the first person to study aesthetics in
chess experimentally. He derived eight principles of beauty in the game from the
judgement of experts (i.e. 30 players with an official Elo rating of over 2000) by
showing them pairs of positions and asking them to select the more beautiful solution
(Margulies, 1977). The Elo rating (see Appendix B), while widely used, is not
necessarily the best or most accurate measure of performance in the game (Donninger,
2003; Lopatka and Dzielinski, 2007; Elo, 2008). The eight identified ‘principles of
beauty’ derived by Margulies are as follows.
1. Successfully violate heuristics.
2. Use the weakest piece possible.
3. Use all of the piece’s power.
4. Give more aesthetic weight to critical squares.
5. Use one giant piece in place of several minor ones.
6. Employ themes.
7. Avoid bland stereotypy.
8. Neither strangeness nor difficulty produces beauty.
The 1st principle involves making a move that goes against basic principles of ‘good
practice’ in chess. If the objective (e.g. checkmate, win material) is achieved despite
such a manoeuvre (e.g. leaving a piece exposed to capture), the move is considered a
successful heuristic violation. The 2nd principle is related to economy. The queen and
rook for example, have similar capabilities along ranks and files on the board. If a rook
22
is sufficient for the task, the solution is considered more beautiful than using a queen
since the former is a weaker piece.
The 3rd principle refers to the distance a particular piece travels. A piece’s mobility – the
number of squares it controls – is a good reflection of its power. Hence, a piece moving
a greater distance across the board is more beautiful because less of its power is wasted.
The 4th principle places emphasis on the piece most involved in the objective. For
example, in a checkmate situation, the piece delivering mate would matter more,
aesthetically, than the one that moved; assuming they were not the same piece (like in a
‘discovered mate’, see section 5.5).
The 5th principle was tested using imaginary pieces not in the original piece set. It was
found that experts preferred positions where all the necessary resources required for the
task were concentrated into one powerful piece, instead of several weaker ones. This
principle is therefore also related to economy or efficiency. The 6th principle, i.e. using
chess themes (e.g. the pin, see section 5.2), is also important aesthetically. Margulies
determined that the more prominent a theme was in a solution, the more beautiful the
solution was considered to be. However, the themes had to be relevant or important to
chess.
The 7th principle implies originality and favours rare positions over common ones, but
the 8th principle seemingly contradicts it. Margulies found that highly unlikely positions
did not lead to judgements of beauty - the experts were actually equally divided between
them and common positions - and neither did solutions which were difficult to find. As
Margulies himself concluded, the 8th principle is rather a restriction of the 7th than its
contradiction. Rarity or originality is favoured aesthetically as long as it is not too
23
difficult to solve, or improbable, i.e. from the viewpoint of its likelihood of occurring in
a real game.
Margulies also questioned intermediate and novice players and found that the majority
of them (a higher proportion in the former) concurred with the experts as to which
solutions were more beautiful. This further supports Lasker’s contention that only
understanding - not mastery - is a prerequisite to appreciating beauty in the game
(Lasker, 1960; Belov et al., 1996). Margulies found that beautiful moves were often also
the most effective ones.
Margulies essentially identified through experimentation with experts many tangible
constituents or elements of aesthetics in the game of chess, and employed game metrics,
similar in some ways to Wilson. For example, pieces were evaluated according to their
relative values (see subsection 3.5.2(a)) and distance was measured in squares bound
only by the limits of the chessboard. The positions Margulies used in his experiments
were restricted to single moves to avoid ambiguity when interpreting the underlying
principle. However, he proposed no model or formalizations for aesthetics even though
the elements were clear. This is probably because his main intention was to investigate
‘traditional’ aesthetic principles (e.g. economy, elegance, novelty) outside the domain of
chess. The game was simply a convenient place to experiment. He found that chess only
confirmed, rather than provided more insight, into the traditional principles. His derived
principles of beauty in the game, though not necessarily a conclusive set (Fine, 1978),
are nevertheless valuable to the research presented in this thesis.
24
2.4 Computer Chess Problem Composition
Schlosser (1988, 1991), in his approach to computer chess problem composition, built
on related work (van den Herik and Herschberg, 1985; van den Herik et al., 1988) that
had been done with regard to chess endgame databases. He outlined three steps that
were required for the process. The model is as follows.
1.Construct a complete database.
2.Eliminate all ‘incorrect’ positions.
3.Select true chess problems.
Similar ideas were later used to compose problems in Tsume-Shogi, a Japanese game
not unlike chess (Hirose et al., 1997; Watanabe et al., 2000). The method was an
improvement over using a random algorithm (Noshita, 1996). A ‘reverse method’ has
recently been proposed (Horiyama et al., 2008) but is not directly applicable to chess
due to certain differences between the games. In Tsume-Shogi, each move of the
attacker must be a checking move; in a ‘shogimate’, only one solution exists; unlike
chess, where a mate might have more than one solution.
The 1st step was restricted to (a database of) endgame positions with a few pieces
because the number of possible positions increases exponentially with the pieces,
making computation unfeasible (Stiller, 1995). Complete databases, tablebases or
‘oracles’ as they are known are designed through retrograde analysis. This involves
starting with say, a checkmate position (that has a specific game state, i.e. ‘won’) and
working one ply or half-move backwards (Thompson, 1986). The process is repeated
25
until a seemingly uncertain position can be shown to lead to a checkmate in the shortest
number of moves and against any defence.
A complete database would therefore include all possible positions of a certain set or
number of pieces, and their inevitable result (win, loss or draw) in a given number of
moves. Presently, a complete database of 6-piece endgames including the two kings has
been achieved (Thompson, 1996). Seven pieces is estimated to be possible by the year
2015 (Hurd and Haworth, 2006). Such databases are also possible in other variations of
the game (Fang, 2006). Efforts have been made to reduce the size of tablebases but they
are still generally quite large with sizes running into gigabytes for just 5 and 6 pieces
(Thompson, 1996; Heinz, 1999a; Nalimov et al., 2000). Given the ‘omniscience’ of
such databases, they are useful for developing learning approaches in the game
(Sadikov and Bratko, 2006).
The 2nd step involves eliminating ‘incorrect’ positions from the standpoint of
composition conventions. Most conventions are not very difficult to formalize and this
step helps to reduce, significantly, the number of positions found that inevitably lead to
checkmate (given orthodox problems). The 3rd step is where aesthetics is considered and
requires the intervention of human composers. Schlosser states the following.
“A computer, however, is not capable of composing like a human being. Creating a new
chess problem according to a given theme, which is the really creative part of a
composer's work, remains to be done by man.” (Schlosser, 1988)
26
“Formally, all positions left after step 2 are correct chess problems. To choose the
‘best’ ones from the potentially large set of correct positions, the imagination and
experience of a (human) expert is needed. According to the criteria of chess
composition, he selects what is new, artistically or aesthetically. There is still no way to
assign this task to a computer. An analogous situation exists in music composition or
painting.” (Schlosser, 1991)
Here, he acknowledges the importance of aesthetics and chess themes which typically
require the experience and expertise of a human composer. Schlosser implied that an
analogous situation exists in music and art, but these domains are more culturally
dependent, and have fewer discrete and computationally amenable components than
zero-sum perfect information games. Even so, computational models which address
aesthetics (in varying degrees) in those domains have since been developed with
reasonable success (Machado and Cardoso, 1998; Golub, 2000; Cope, 2001; Manaris et
al., 2002a, 2002b; Datta et al., 2006). These are beyond the scope of this thesis but their
methods are briefly discussed in section 2.8.
On its own, Schlosser’s first two steps were capable of finding forced checkmates that
were hard to solve by humans and occasionally featured interesting themes. This is to be
expected since themes or tactics are an integral part of how the game is won. The
problem, therefore, is in getting computers to recognize the aesthetics of a composition
or game for its own sake because somewhere in a massive tablebase certainly lie even
the most beautiful compositions that humans could conceive with those pieces and
would appreciate.
27
The commercially available program ‘ChessExplorer’ uses a similar two step process to
create chess problems of the mate-in-2 and 3 varieties. However, its second step does
not filter them using any criteria except checking for a forced mate with only one
solution. Hence, the ‘created’ problems are usually not attractive; this is still evident in
the latest version of the program i.e. v6.11 (Nowakowski, 2005, 2008).
Schlosser’s model provides a clever way to emulate creativity in composing through the
use of brute-force searching but still relies on human intervention for the aesthetics
component. He therefore separates aesthetics from composition convention and
concedes to the limitation of his approach. The author hopes that this research will
address the aesthetics component in a way that can be incorporated into models like the
one proposed by Schlosser.
The automation of problem composition can then be improved so it does not need to
rely on human intervention as much or at all (more recent work is discussed in section
2.7). Schlosser’s model limits the scope of automatic composition to orthodox problems
and what are possible using available endgame databases. Any reasonable aesthetics
model would also need to be limited in this way to be consistent with available
information, and feasible in terms of required computing power. In summary,
Schlosser’s approach clearly identifies the gap a computational aesthetics model would
fill in this context.
2.5 Elements of Beauty Classified
One of the recent books that address aesthetics in chess is, ‘Secrets of Spectacular
Chess’ (Levitt and Friedgood, 1995, 2008). The book is currently in its expanded 2nd
28
edition. In it, the authors – a chess grandmaster and international master of problem
solving, respectively – classify four elements of chess beauty namely paradox, depth,
geometry and flow. The book features a section entitled, ‘The Importance of Chess
Aesthetics’ and lists several reasons to support that contention. These include pleasure,
cultural or artistic value, educational and practical value.
Aesthetics in the game gives pleasure to a person and his life is considered to be more
meaningful than one who is unable to derive the same pleasure from it. Cultural or
artistic value is compared to paintings which exhibit the skill and genius of their artists.
In terms of education, good problems and pretty studies (a form of chess composition,
see ‘endgame study’ in Appendix B) are seen as an excellent teaching tool with
surprising solutions that can capture the imagination of those learning the game,
especially children. Finally, beautiful compositions and games have practical value in
actually improving a person’s – even a master’s - quality of play because they are full of
effective and original ideas.
Levitt and Friedgood write that virtually all world class players (e.g. Kasparov,
Botvinnik and Lasker) have an interest in the aesthetic aspect of chess and that it has
helped in their development. Returning to their elements of chess beauty, ‘paradox’
means a violation of heuristics or doing something that one is not usually supposed to
do, e.g. leaving a piece in a position to be captured. It is paradoxical because the move
wins despite going against general ‘good practice’. Successful heuristic violation (the
first principle of beauty derived by Margulies, see section 2.3) comes under this.
‘Depth’ refers to the point of the key move being obscured or unclear at the beginning
but realized later. It is the sort of foresight in a move that does not make much sense at
29
first and is not necessarily paradoxical, but makes perfect sense by the end of the
combination.
‘Geometry’ is the chance or planned formation of shapes on the chessboard that
resemble say, alphabets. While this is very rare in real games, some compositions
feature it. Geometry also includes other visual effects on the board such as symmetry
and the relevance of particular lines (i.e. ranks, files, diagonals) to a solution. ‘Flow’
describes a move sequence that is basically forced instead of complicated with many
side variations. Flow is therefore more common in real games than in compositions
where side variations may even be laudable. Themes – though not explicitly classified
as an element of beauty – are treated as a given rather than examined individually. This
is probably because not all implementations of chess themes are noteworthy from an
aesthetics standpoint even though most examples of exemplary beauty tend to feature
some theme or other.
Levitt and Friedgood succeed in presenting to a modern audience the aesthetic aspect of
chess using contemporary examples and styles of play. This is important because in the
computer age, it is sometimes thought that little is now left to the imagination,
especially for a zero-sum perfect information game. Their examples (of beauty in the
game) illustrate that there are still useful tactics and strategies that our current
computational approach to playing is unable or slow to recognize. This remains true
even today. In other words, we are still better at solving certain problems in the game
using our creativity than computers, despite their brute-force approach. Levitt and
Friedgood, however, do not themselves propose any models or formalizations for
aesthetics. Yet, their broad classification of the elements of beauty and well-chosen
30
examples of the finer aspects within them elucidate the principles of aesthetics proposed
by others (e.g. Margulies, though he is not cited) and extend them even further.
For instance, some of their examples suggest the importance of mobility (the number of
squares controlled by a piece in a particular position) as an additional property for
aesthetic computation (in addition to piece value and distance). Other examples
demonstrate the wide variety of possibilities within individual chess themes, which
Margulies identified as his 6th principle but only briefly explained. Similar to Lasker,
their contributions stem from experience rather than experiment (although it is notable
they had the distinct advantage of computer analysis) and they make no significant
distinction between compositions and real games in terms of aesthetics. They also do
not suggest a way of thinking about aesthetics as an independent component that is not
necessarily exclusive to composition conventions or real games. Nevertheless, their
examination of aesthetics in the game contributes significantly to the literature on the
subject.
2.6 Beauty Heuristics in a Game Engine
Aesthetics in chess has also been applied as a viable alternative to traditional game-
playing heuristics. Walls (1997) proposed using some of the principles of beauty
derived by Margulies (see section 2.3) in a chess engine to see if it performed better
than one that used standard heuristics. It was found that the engine using beauty
heuristics was 25% faster and needed to analyze 33% fewer nodes (i.e. positions) than
the one using standard heuristics, for solving direct-mate chess problems between 2 and
5 moves long.
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There are two important implications of Walls’ research. First, it supported his position
that humans make good moves in chess (at least in part) based on their ‘sense of
beauty’. Second, his findings also suggest a correlation between effectiveness and
aesthetics in the game. This is similar to using beauty as a ‘measurement of
performance’, and it has been suggested not only in chess but also in economics
(Katsenelinboigen, 1990, 1997). It is not known, however, if the results obtained by
Walls remain true in a full game. He did not apply all of the principles derived by
Margulies because not all of them were applicable to the scope of his research.
The first principle (see section 2.3), ‘successfully violate heuristics’ was adopted but
excluded standard heuristics that did not apply to mating problems. These included
those that (the violation of which) made it difficult to find the forced mate. The second,
third and fourth principles were summarized as ‘do not waste any power’. They
encapsulate traditional aesthetic principles such as economy, parsimony and simplicity.
Walls modified the fifth principle to ‘use all of the pieces’ because the original one used
imaginary pieces and could also be interpreted as wasting less power (or using all
available pieces). He eventually rejected this modified principle due to computational
overhead. Themes were not included for the same reason and because they were thought
to distract from finding the quickest solution to checkmate. The seventh and eighth
principles by Margulies were also not included.
The heuristics implemented in the standard game engine were limited to those
applicable to mating problems and included the following.
1. Place the enemy king in check.
2. Attack the squares surrounding the enemy king.
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3. Sacrifice pieces if they lead to checkmate.
4. Gain enemy material.
The ‘beauty’ version of the engine included the first three standard game engine
heuristics and the following additional ones.
1. Violate the ‘gain enemy material’ heuristic.
2. Use the weakest piece possible (to check).
3. Use all of the piece’s power.
4. Give aesthetic weight to the critical piece.
The first additional heuristic was limited to violating just the 4th standard heuristic
because violation of any of the others would impede finding the checkmate solution.
Walls implemented these beauty heuristics in a rather rudimentary manner. For
example, in order to encourage the engine into making sacrificial moves, the standard
evaluation function which calculates the material balance of a position (typical in all
chess programs) was disabled. Weaker pieces (pertaining to the second additional
heuristic) were determined using the standard Shannon values but limited to ‘checking’
moves. The third additional heuristic (i.e. use all of the piece’s power) involved
counting the number of squares made by the checking piece. There was no difference
therefore, between a queen moving across 5 squares and a bishop moving the same
distance. While this speeds computation, it may not correlate well with human aesthetic
perception of the manoeuvres since they involve pieces of different value.
The fourth additional heuristic awarded extra points to the move based on the number of
higher valued pieces in the piece set than the one performing check. Weaker pieces
33
therefore, were considered more critical than stronger ones. In general, Walls may have
compromised his experiments to a degree by oversimplifying the aesthetic principles he
used. The reason he did this was because the main focus was not aesthetics itself but an
improvement in game-playing heuristics which requires fast computation for efficient
searching.
A complex representation or formalization of the aesthetic principles, however
necessary, would have slowed down the beauty heuristics engine considerably. It is
difficult to say if any realistic measure or identification of aesthetics was attained
through his experiments. Initial tests with human players on the aesthetics component
alone would have established this. Even so, Walls demonstrated - with some degree of
success - a computational implementation of chess aesthetic evaluation that can
potentially improve game-playing heuristics.
2.7 Computational Improvement of Chess Problems
Some of the relatively recent research in the area has sought to improve the composing
ability of computers with regard to chess because unlike playing, computers are quite
poor in problem composition. The Improver of Chess Problems (ICP) was presented as
a model to improve the quality of two-move mate problems (HaCohen-Kerner et al.
1999). A significant proportion of the knowledge required to evaluate the quality of
compositions was formalized for this purpose through consultation with two
international masters of chess composition. Based on the model, a chess problem is
typically put through several ‘transformations’ in order to improve it. These
transformations include the following.
34
1. Deletion, addition or replacement of a piece on the board.
2. Transfer of a piece to another square within the same rank or file.
3. Transfer of a set of pieces using a particular movement (e.g. 3 files to
the right).
Each transformation is tried on a specific piece on the board (in a sequence of
importance) and applied only if the new position satisfies the three criteria mentioned
below.
1. It is legal.
2. It is a two-mover with only one key move.
3. It has a higher quality score.
The new position does not need to include the themes of the original problem or the best
quality score of the best transformation found so far. Thematic considerations were
considered restrictive to the number of improvements possible while weaker initial
transformations were seen as possibly leading to better overall improvement. The
quality evaluation function they used is shown below. V denotes the value function, Ti
the set of all themes in the position, Bj the set of all bonuses granted and Pk the set of all
penalties imposed.
( ) ( ) ( )severe deficiency
otherwise
0
m
i j ki j k
qV T V B V P
= + −∑ ∑ ∑
Ten themes common to compositions (e.g. direct battery, Grimshaw) were attributed
relative but fixed values ranging from 10 to 45. Bonuses included desirable practices in
35
composition such as placing the black king in the centre of the board (10 points) and
certain manoeuvres like sacrifices in the key move (5 × piece’s value). Penalties ranged
from severe deficiencies (e.g. illegal position, not a mate-in-2) to smaller things such as
a check (-50 points) or pinning a black piece (5 × piece’s value) in the key move.
The ICP was tested on 36 orthodox mate-in-2 chess problems taken from composition
books and managed to improve 10 or 27.7% of them. Eight of the ten were improved
after a single transformation and the remainder after two transformations. Mate-in-2
miniatures (no more than 7 pieces on the board) were used so that improvements could
be achieved within a reasonable amount of time. The low proportion of improvements
obtained from the 36 problems was attributed to the fact that these were known
compositions and mostly already optimized.
Aesthetics was not explicitly accounted for in the ICP model even though some of the
knowledge for evaluating the ‘quality’ of compositions included certain principles of
aesthetics such as themes and sacrifices. There are two significant issues. First, the
relative (fixed) values of themes, bonuses and penalties seemed to have been
determined arbitrarily by just two master composers, including the final determination
of perceived improvement over the original problems in the experiment. It is
noteworthy that the flow of information from expert to non-expert in complex domains
like chess often results in a bottleneck which affects the quality of the knowledge
formalized (Michie, 1986; Guid et al., 2008). Aesthetically, this approach does not
account for varying configurations of particular themes (e.g. using different pieces in
different places for a similar purpose) and the perception of composers in general; the
majority of whom are not masters.
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Second, this model is not easily applicable to real chess games where aesthetics is also
perceived. The ICP was designed specifically with chess problems in mind.
Nevertheless, the ICP is an improvement over Schlosser’s model (see section 2.4)
because there is an attempt to deal with the aesthetics component, albeit in a way that
conflates it with convention (comparable to Wilson’s system, see section 2.2).
An improved model called, ‘Chess Composer’ used a similar approach to the ICP but
had fewer types of transformations (Fainshtein and HaCohen-Kerner, 2006a, 2006b).
This reduced the branching factor and increased the depth of applied transformations.
Chess Composer used brute-force searching to find a global maximum and did not
suffer from ICP’s limitations in terms of pruning the search tree, 1) wherever the
position was not legal, 2) not a two-mover with one key or, 3) with a lower quality score
than the original problem. Much of the domain knowledge was taken from the ICP
model with some additions, but the quality function remained the same.
Chess Composer was tested on 100 mate-in-2 chess problems and managed to improve
the quality of 97 of them. Despite its relative slowness, the model’s higher success rate
can be attributed to using a better search technique and greater depth of transformations
(3 levels instead of 2). Most of the improvements were, in fact, achieved after various
sequences of three transformations. The authors recognized Chess Composer’s
limitations in terms of aesthetics and state the following, before alluding to the use of
ICP’s formalized knowledge to compensate.
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“The concept of a ‘high-quality mate problem’ in chess is hard to define, especially if
an automatic program is involved. It is not simple to define concepts such as beauty,
originality, uniqueness of the solution, and difficulty to solve.”
(Fainshtein and HaCohen-Kerner, 2006a)
Both the ICP and Chess Composer models are practical methods of ‘composing’ or at
least improving existing chess problems within a limited scope (e.g. mate-in-2) from the
standpoint of composition conventions. Longer problems are possible but would require
more effective search techniques in order to be computationally feasible (Fainshtein and
HaCohen-Kerner, 2006b). They admit that improving the quality function might
contribute to that (Fainshtein and HaCohen-Kerner, 2006a). In general, Chess
Composer improves more in terms of performance rather than technique, when
compared to the ICP, and does not look any deeper into the question of properly
accounting for aesthetics.
The conflation between composition convention, and aesthetics, still exists and as a
result the latter is improperly accounted for. The attribution of fixed but relative values
to themes and conventions with the aid of masters in composition (for both the ICP and
later supplemented in Chess Composer) is a step closer toward complete automation of
the process but nonetheless fails to account for aesthetic variety within each theme and
convention. Even though some conventions are described using simple formulas, e.g.
bonus for X number of pieces on the board = 3 × (18-X), the constants used are not
explained and therefore presumably have little basis in chess literature.
Given the brute-force searching required in both of these models, complexity with
regard to aesthetics was perhaps rightfully avoided. The research presented in this
38
thesis, however, deals exclusively with aesthetics and a model which should enable
computers to recognize it in the game like humans do. Since brute-force searching for
larger purposes (e.g. automatic composition) is beyond the scope of this thesis,
formalization of the available knowledge we have on chess aesthetics need not be
compromised. This means that the variety of configurations possible within the
established aesthetic principles and themes can be better accounted for.
2.8 A Look at Methodologies Used in Other Domains
Even though the methodologies used in other domains are not quite applicable to zero-
sum perfect information games (the domains are of different natures), this section
presents a brief discussion of a select few for a general context. One of the earliest
attempts to formalize aesthetics is the mathematician Birkhoff’s model of, M = O/C
where M = aesthetic value, O = order and C = complexity (Birkhoff, 1933; Scha and
Bod, 1993). ‘Order’ was identified with factors like symmetry and repetition, and
‘complexity’ with the amount of effort required to attend to a pattern and assimilate it.
These enabled similarities and relations between elements to be discerned using
numerical values. Even though Birkhoff specified procedures for attributing such values
to certain things such as polygons, vase outlines, melodies and lines of verse, his model
was not successful mainly due to oversimplification. Maximum aesthetic satisfaction
was simply not obtained using the most order and least complexity in patterns, as his
model suggested. Experimental results proved this (Berlyne, 1972).
Later attempts have limited such models to just one domain and identified the measures
used to factors strictly within that domain. Machado and Cardoso (1998) for example,
39
equated the aesthetic value of artwork (i.e. images) as IC/PC where IC = image
complexity and PC = processing complexity. IC was estimated as the amount of error
during lossy compression (e.g. JPEG) divided by the compression ratio. PC was
estimated using fractal compression because they had reason to believe it was closer to
how humans process images in their mind. Experimental results with users were
encouraging.
Datta et al. (2006) derived aesthetic factors for photographs from rules of thumb in
photography, common intuition, and observed trends in ratings. These were compared
against aesthetic ratings of over 3,000 random photos by members of an online photo
community to identify the features that correlated well (Physorg, 2006). A classification
and regression model was then built using a subset of relevant features. Regression
models can be used to obtain absolute quantitative results for aesthetic features whereas
classification models work better for qualitative results (i.e. high and low thresholds).
They found the latter to be more suitable because it was difficult to differentiate
between the aesthetics of photos that scored closely.
Manaris et al. (2002a, 2002b) used discrete representations (e.g. frequency of notes,
intervals) of particular attributes in music (e.g. pitch, volume) to recognize beautiful
compositions. They found that the aspects of beauty in music may be algorithmically
classifiable and identifiable based on an experiment which compared ‘quality’ MIDI
renderings of musical pieces (by known composers from various music genres) against
random pieces (i.e. noise) by testing for conformity with the Zipf–Mandelbrot Law.
This law essentially states that phenomena generated by complex social and natural
systems tend to follow a statistically predictable structure. Examples include human
40
language and music. They acknowledge, however, that their statistical approach has
limitations; notably that minor changes to a particular musical piece (e.g. one note to
another) may be statistically insignificant but highly significant (aesthetically) to the
listener.
A more recent approach combined statistical, connectionist and evolutionary
components with the assumption that popularity of music correlates with aesthetics
(Manaris et al., 2007). There is potential in evolutionary algorithms for both the
aesthetics of music and art (similar to biological evolution) but it challenges what we
even classify as ‘art’ because ordinarily it is ‘created’ by humans, not evolved
(McCormack, 2006). Methods used for larger purposes such as composing and
classifying musical styles (Golub, 2000; Cope, 2001; Miranda et al., 2007) often do not
address aesthetics directly so they are not discussed here.
The computational approach to aesthetics in other domains is generally problematic
because aesthetic features or principles are often loosely defined and culturally
dependent. This means that the experts themselves find it difficult to be specific enough
about the constituents of beauty within a particular domain. Researchers therefore, find
it easier to extract and weight features on their own through the analysis of domain-
related resources such as photographs, music compositions or prose. There are many
ways this analysis can be done and justified. The extracted features may or may not
have anything to do with aesthetics in the domain but are then usually tested against
human perception to identify which ones are most likely related to beauty.
Computer systems (usually after some training) can then use formalizations of those
features to recognize aesthetics (to a limited degree) in other objects within that domain.
41
It is difficult to apply features or methods from one domain to another because in many
cases, they are significantly dissimilar and perceived through different senses. The
objects themselves (e.g. literature and photographs) may use different ‘channels’ even
in cases where the modality or sensory organ (e.g. the eye) is the same (Vaughan,
2006). For that matter, the aesthetic criteria used to evaluate novels are likely different
from poetry or plays, despite the same channel and modality (Weyhrauch, 1997).
There are two main advantages - in terms of aesthetic evaluation - to a zero-sum perfect
information game like chess. First, it is theoretically finite and particularly amenable to
computation. Second, because the rules and pieces are the same anywhere (at least for
international chess), it is not subject to cultural influence and marked difference of
opinion, even aesthetically. While cultural influence or upbringing may affect the
general style of a person’s play, the objectives of the game are the same and must be
adhered to. Losing a queen unnecessarily for example, is inadvisable regardless of
where you are from.
This makes aesthetic principles within that theoretically finite domain arguably more
reliable and consistent than in any other. The typical approach of performing regression
analysis to determine aesthetic weights, and neural networks to train the system, is
possible; but there is no inherent subjectivity in chess that calls for it. Reliable,
aesthetically-rated chess combinations required for such an approach are also scarce.
Just as chess programs tend to simulate the abstract intelligence of human playing
ability through the use of discrete brute-force techniques, it is possible that human
aesthetic perception in the game can also be simulated using an analogous approach. It
may, in fact, be the most suitable one.
42
Given that chess literature has sufficient information relating specifically to the
aesthetics of the game, it was not deemed necessary to conduct further experiments to
derive more principles. Additionally, the resources necessary for that (e.g. a large
number of master players) were not available to the author. The approach taken in this
research can therefore be considered unique compared to how computational aesthetics
is addressed in other domains. The details are in the following chapter.
2.9 Chapter Summary
Research into chess aesthetics started with the recognition by master players and
composers that it is a significant aspect or dimension to the game. They showed that it is
not limited to compositions and also extends to real games. Since aesthetics is more
prominent in compositions, it is often conflated with composition convention, and not
treated separately in a way that would make computational application of it easier. This
is especially true when looking at real games where composition conventions do not
necessarily apply, yet beauty is also perceived. In the case of orthodox problems, there
is no difference between compositions and tournament games except in the way the
positions are configured to reflect certain principles and themes.
Attempts to classify and quantify aesthetics have been made through the systematic
identification of principles and themes that are present in what players consider
beautiful positions and combinations. These have been essential to the development of
models and formalizations which try to account for aesthetics as a by-product while
focusing on conventions that are more easily defined and recognized on the board.
Applications of this approach therefore focus on automatic composition of problems in
general (within a limited scope) but produce relatively poor results in comparison to
43
human composers. Part of the reason lies in the rudimentary adaptation of the
established aesthetic principles and themes by using typically fixed, but relative values;
even though this is sometimes done through consultation with master composers.
The variety of configurations possible in each principle and theme (which is what
makes a combination beautiful) is greater than what can be realistically accounted for
with fixed values, even relative ones. The number and suitability of principles and
themes identified that apply to both domains (i.e. compositions and real games) are
equally important. This is a compromise which researchers have often made because
their models rely on brute-force searching that works best by minimizing computational
load. They also have no conceptual framework for aesthetics that enables it to be
described independently of composition conventions. In most models, aesthetics is not
explicitly accounted for so the quality functions designed to evaluate it often suffer from
oversimplification. The principles and themes typically used are also limited and often
restricted to just a selection within the domain of compositions, which neglects the
portion of aesthetics that also exists in real games.
The repercussions of all this are significant because an inadequate or poor account of
aesthetics will affect any subsequent application of such a model. This might include the
following.
1. Versatile chess database search engines (e.g. that are able to
locate beautiful games and combinations automatically).
2. Programs that aid judges of composition tournaments and
brilliancy prizes.
3. Chess program personality modules (for more human-style
play).
44
4. Game-playing engines (based on beauty heuristics).
5. Automatic chess annotators (Guid et al., 2008).
The issues apply not only to international chess but possibly hundreds of other chess
variants and similar zero-sum perfect information games with an aesthetic dimension
that could benefit from such a model. The next chapter presents the research
methodology and proposed aesthetics model.
45
CHAPTER 3: METHODOLOGY – Aesthetics in the Game
3.0 Components of the Research
This chapter details the aesthetics model (for the game of chess) proposed in this thesis.
The model was designed to answer the research questions summarized in section 1.6.
Section 3.1 first outlines all the components of the model. The first five components are
presented and explained, in sections 3.2 through 3.6. Component 6 is divided into
chapters 4 (aesthetic principles) and 5 (themes). The scope of analysis for this research
is explained in section 3.7 and the points of evaluation (for aesthetics in a move
combination) in section 3.8. Experimental validation of the model (based on the scope)
and related procedures, is addressed separately and presented in chapter 6. The term
‘formalization’ (noun) is used to refer to an evaluation function expressed as a
mathematical formula. A glossary of chess terms is provided in Appendix B for
reference.
3.1 The Proposed Model of Aesthetics
The model of aesthetics proposed in this thesis consists of the following components.
1. A conceptual framework for aesthetics in the game.
2. An examination of aesthetics in the game.
3. A selection of aesthetic principles and themes.
4. A formula for the cumulative aesthetic assessment in a move
sequence or combination.
46
5. A general methodology for developing formalizations for
aesthetic principles and themes.
6. The actual and detailed formalizations for the selected aesthetic
principles and themes.
In principle, this model makes aesthetic evaluation in chess computationally feasible.
3.2 A Conceptual Framework for Aesthetics in the Game
In order to approach the question of aesthetics in the game of chess, a workable
definition and conceptual framework is desirable. ‘Conceptual framework’ here refers
to a way of thinking about aesthetics (in the game) that can guide proper investigation
of it. A significant problem with existing methods of dealing with aesthetics is that they
do not address it exclusively or directly. Rather, it is addressed indirectly by conflating
the term with ‘composition conventions’, or assumed to be an unintentional (e.g.
emergent) result of said conventions (Wilson, 1978; Schlosser, 1988, 1991; HaCohen-
Kerner et al. 1999; Fainshtein and HaCohen-Kerner, 2006a, 2006b). This makes it
difficult to formalize the knowledge we have about aesthetics, specifically.
Aesthetics in the game can be loosely defined as, ‘what players perceive to be beautiful
in it’ (Lasker, 1960; Levitt and Friedgood, 2008). The word ‘players’ here also includes
problem composers (who must also know how to play the game). This should be
clarified because composers and players are sometimes thought of as two distinct
groups with different ideas about aesthetics in the game, i.e. they find different things
beautiful. Technically, there is no basis (inherent to the game itself) for any such
47
difference; especially when the pieces, board, rules, objective and move length of the
combination being evaluated are the same.
Said definition does not, however, extend to the many different designs of chess sets
which are beautiful in their own right (Schafroth, 2002). It is limited to the means (e.g.
principles and themes) through which ideas can be expressed using any set. Since
beauty is perceived in both real games and compositions, it stands to reason that
aesthetics can be evaluated as something separate from composition conventions even
though the two are not necessarily mutually exclusive. Aesthetics is generally
considered to be more prominent in compositions because composers have the luxury of
time, and advantage of design, in contrast to real games where competitiveness or
winning is the primary objective (Lord, 1984-5; Humble, 1993, 1995; Ravilious, 1994).
In real games, aesthetic moves usually occur as a result of sound play and chance
(Linder, 1981; Damsky, 2002; Sukhin, 2007). A grave error by one player can
sometimes facilitate a beautiful combination by the opponent.
‘Orthodox problems’ are similar to real games in terms of rules and have the additional
requirement that they must be a possible occurrence in a real game (Rice, 1997). Hence,
from an aesthetics standpoint, the only difference between these compositions and real
games is in terms of their perceived beauty. Other purely strategic or perceptual
differences in patterns between them (perhaps prosaic in themselves) are not necessarily
exclusive of this beauty or likely to exclude it. For real games, no distinction is made
between those achievable through legal moves and ‘reasonable’ ones likely to be played
by masters (de Groot and Gobet, 1996). In compositions, the term ‘beauty’ is often
interchangeable with convention but in real games, composition conventions need not
be followed, even though they may be present unintentionally. Therefore, aesthetics
48
(exclusively) in the game can be conceptualized as the common ground between
compositions and real games in terms of what is perceived by players to be beautiful.
This might include some conventions but not all of them. Since all orthodox problems
must be possible occurrences in a real game, compositions of this nature are technically
a subset of real games. Not all positions in real games are considered compositions. For
instance, those that do not have any reasonable stipulation such as ‘mate-in-3’ or ‘white
to play and win’ are not included.
Aesthetics is perceived in both the subdomain of orthodox compositions and outside of
it (e.g. in real games that are not governed by composition conventions), though more
prominent in the former. Figure 3.1 illustrates the concept. ‘Compositions’ refers to the
orthodox variety.
Figure 3.1 Concept of Aesthetics in Chess
The diagram represents all positions that could occur in the game of chess.
Compositions (including their adherence to convention) are necessarily positions that
could legally occur in a real game. Aesthetics or beauty is more prominent in
compositions (less in real games) and is represented by the chequered area. For
explanatory purposes, this indicates positions that have achieved a hypothetical
Compositions
Real Games
Aesthetics
49
minimum threshold of beauty that distinguishes them from the rest. There are typically
more of these in compositions than real games but they exist nonetheless, in both
domains. Even though orthodox compositions are technically a subdomain of real
games, both are referred to as separate ‘domains’ for the purposes of this thesis. The
area of aesthetics that overlaps with compositions happens to share in their conventions
whereas the area that overlaps with real games does not, necessarily. The chequered
area (as it pertains to the chosen scope) is therefore what this research is attempting to
capture computationally because it is appreciated and often sought after by humans
(players and composers alike).
Thinking about aesthetics in this way makes it easily applicable to both real games and
compositions with less confusion than if it were conflated with composition
conventions. This would mean that automatic problem composers (i.e. computer
programs that compose) for example, would have to take into account not only
convention, but also the aesthetic component that overlaps with real games.
3.3 An Examination of Aesthetics
The following subsections compare composition conventions and aesthetic (i.e.
brilliancy) characteristics in real games, against the general principles of aesthetics
found in chess literature. A reasonable ‘common ground’ is then identified for
computational purposes. Note that, when referring to a player by the colour of his
pieces, the first letter is usually capitalized (e.g. ‘White’, ‘Black’).
50
3.3.1 Composition Conventions
Composition conventions are typically assumed or understood to encapsulate aesthetics
and consist mainly of the following (Lipton, 1965; Howard, 1967; Rice, 1997; Polgar
and Pandolfini, 2006). The chess composition should:
1. illustrate some particular powers of the chessmen in their interaction with
one another;
2. possess a solution that is difficult rather than easy;
3. contain no unnecessary moves to illustrate a theme;
4. contain more variety in the defences available to the opposing side (Black)
but they must be related to the thematic content of the problem;
5. possess complexity of variations;
6. have White move first and mate Black;
7. have a starting position that absolutely must be possible to achieve in a real
game, however improbable;
8. contain only pieces present on the board at the beginning of the game, i.e.
no more than 1 queen, 2 rooks, 2 bishops (of opposite colour squares), 2
knights and naturally 8 pawns; however, pawns may be promoted to any
piece in the actual solution;
9. not allow ‘en passant’ moves unless they take place as legitimate moves in
the solution, or have them functioning as a key (the first move) unless
retrograde analysis shows Black’s last move to permit it;
10. avoid castling moves because it cannot be proved legal;
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11. have a key move that appears aimless or inconspicuous, i.e. violates chess
heuristics, meaning that strong moves (e.g. checking, captures, limiting the
mobility of Black) are undesirable;
12. possess more moves in the solution that are also of the ‘quiet’ type (i.e.
inconspicuous);
13. possess only one unique key move that will solve the problem, otherwise it
is ‘cooked’;
14. have a definite solution in the stipulated number of moves immune to any
unexpected defences by Black;
15. preferably not contain ‘duals’ or ‘triples’ but this cannot be entirely
eliminated from compositions, so the issue is usually explored in greater
detail and may vary depending on the judge;
16. feature economy, i.e. a good relation between the number of chessmen used
and the results obtained (based on complexity or variety in lines of play); a
problem is considered uneconomical when the same result could be
obtained with fewer chessmen or less powerful ones, so a piece should be
made to use as much of its power as possible with more emphasis given to
the white forces in this respect;
17. create a deceptive setting for the solver (makes it look like a different theme
is at play) so to lend more satisfaction when the real solution is discovered;
18. not be ‘dressed’ (placing unnecessary pieces to mimic the conditions of a
real game) which used to be the practice of earlier composers but today
interferes with the concept of economy;
19. have the chessmen spaced over the entire board rather than just in one
section, as too many pieces close to each other depict clutter;
52
20. avoid using too many pawns, especially mutually blocking white and black
ones; doubled and tripled pawns are objectionable, except when used
thematically;
21. not place pieces in ‘unnatural’ positions; a skilled composer endeavours to
keep his positions from appearing this way.
Composers, who do not adhere to these conventions, risk their compositions being
rejected in composition tournaments, evaluated unfavourably, or not being published.
However, this does not necessarily mean that ‘unconventional’ compositions are not
beautiful. Some composers do not adhere to certain conventions when they feel it is
justified. Compositions have evolved over the centuries but can presently be classified
into three major groups or ‘schools’ namely the Bohemian school, the Logical school
and the Strategic school (Howard, 1967; Zirkwitz, 1994; Levitt and Friedgood, 2008).
‘Bohemian’ problems focus on model mates. Their main feature is the economical use
of pieces. ‘Logical’ (or New German) problems are structured ones. They focus on
compositions which essentially feature using a foreplan to negate Black’s defence to a
primary plan of attack (see Appendix C, Figure C.1). ‘Strategic’ problems focus on
variations in a composition with emphasis on the number of elements shared among
them (i.e. unity). The conventions listed above usually apply to all schools of
composition.
3.3.2 Brilliancy in Real Games
Real games are considered beautiful or brilliant for similar but fewer reasons than
compositions. These reasons or characteristics also cover a broader meaning. A brilliant
53
combination in a real game should have the characteristics mentioned below (Damsky,
2002). Equivalent descriptions can also be found in other sources (Smirnov, 1925;
Bronstein, 1983; Avni, 1998; Lionnais, 2002; Levitt and Friedgood, 2008).
‘Expediency’ implies effectiveness such that a move achieves something tangible like
checkmate, decisive material gain or forces a draw in a seemingly lost position.
‘Disguise’ occurs when a move (usually the key move) played, does not expose the
solution immediately. In certain endgame positions such as KR vs. K (short for ‘King
and Rook vs. King’, a typical endgame setting), the winning solution is fairly obvious
and mechanical, so these do not qualify (Thanatipanonda, 2008). ‘Sacrifice’ refers to
that of material (e.g. exchanging a more powerful piece for a weaker one), but can also
mean sacrificing piece mobility or other less tangible advantages.
‘Correctness’ simply means that the desired solution should work against any defence.
While this is not always possible in real games, it is usually a plus. ‘Preparation’ is
something that suggests the aesthetics perceived - usually in the form of a particular
move combination (Sukhin, 2007) - was achieved in great part due to the strategic play
preceding it. Under these circumstances, the whole game may be considered beautiful
and awarded a brilliancy prize. In most cases, however, brilliancy can be pinned down
to one combination. Even in the time of Caliph Harûn Al-Rashîd, in Baghdad (circa 800
AD), a chess player unfolding a remarkable combination might have become the
recipient of a fabulous reward (Yalom, 2005).
5. preparation
6. paradox
7. unity
8. originality
1. expediency
2. disguise
3. sacrifice
4. correctness
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‘Paradox’ covers a wide range of things that go against ‘good practice’ in chess, such as
not exposing your king, or not capturing enemy material when it is possible. One some
level, it can also be said to include sacrifices but that is better treated as a separate
characteristic. ‘Unity’ implies that all the pieces (i.e. in their respective moves) worked
together toward a particular, just aim. This also includes different winning variations
that might be possible. Lastly, ‘originality’ refers to something the observer has not seen
before and therefore relies heavily on personal experience.
It is reasonable to assume that no idea in chess is truly original anymore given the
extensive analysis done by computers to date, and the grandmasters working with them.
However, it is unlikely that every possible expression (i.e. piece configuration) of those
ideas has been observed or thought of by humans. Judges usually rely on these
characteristics as a rough guide when awarding brilliancy prizes to real games because
unlike composers, players have little control over how beautiful their games are,
especially in tournaments which typically have time controls and are competitive.
3.3.3 Principles of Aesthetics
Aesthetics can mostly be found in the overlap of composition conventions and brilliancy
characteristics in real games. This is reflected in the principles of aesthetics mentioned
in chess literature, which include the relevant aspects of both convention and brilliancy
that are applicable to either domain. The following is a list of six general aesthetic
principles derived collectively from that resource (Osborne, 1964; Margulies, 1977;
Bronstein, 1983; Lord, 1984-5; Rachels, 1984-5; Humble, 1993, 1998; Ravilious, 1994;
Sukhin, 2007; Levitt and Friedgood, 2008).
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1. Violate heuristics successfully.
2. Win economically.
3. Sacrifice material.
4. Spread out the pieces.
5. Create geometric patterns.
6. Employ chess themes.
The 3rd principle of sacrificing material is not classified as a violation of heuristics
because the latter implies more subtle violations, such as leaving one’s king prone to
check or not capturing enemy material favourably when it is possible. Many of the
composition conventions and brilliancy characteristics listed earlier can be described
(even collectively) using one or more of these aesthetic principles. For example, ‘violate
heuristics successfully’ is generally inclusive of conventions number 11, 12 and 17 (see
subsection 3.3.1) and characteristics number 2 and 6 (see subsection 3.3.2). Table 3.1
shows how the six general aesthetic principles above relate to both composition
conventions and brilliancy in real games.
Table 3.1 General Aesthetic Principles, Conventions and Brilliancy Compared
General Aesthetic Principles
Composition Conventions
(see subsection 3.3.1)
Brilliancy Characteristics
(see subsection 3.3.2) 1 violate heuristics successfully 11, 12, 17 2, 6 2 win economically 3, 8, 16, 18, 19, 20 1, 7 3 sacrifice material 1, 11, 17 3, 6 4 spread out the pieces 18, 19, 20, 21 - 5 create geometric patterns - - 6 employ chess themes 1, 3, 4, 5 5, 7
All of the brilliancy characteristics in real games are encompassed by the aesthetic
principles except ‘correctness’ and ‘originality’ (i.e. 6 out of 8). The former is not a
56
prerequisite to aesthetics because it depends a great deal on detailed post-analysis of the
moves (Belov et al., 1996). In some compositions such as selfmates and helpmates, it is
even considered aesthetically inhibiting; helpmates are the second most popular type of
chess problem after the direct-mate (Friedel, 2002). It was found, for example, that
some beautiful themes can only be illustrated without a forced sequence of moves
(White, 2003). See Appendix C, Figure C.2 for an example of the helpmate (where
Black usually moves first).
A beautiful combination - as perceived by spectators and even the players - performed
in a real game may later turn out to be ‘incorrect’ because the opponent missed a
possible defence, or the winner a more effective attack. While this may diminish its
aesthetic appeal somewhat (but not entirely), the original perception of beauty still
stands in merit of the combination itself (Znosko-Borovsky, 1959; Lasker, 1960). A
typical example is Adolf Anderssen’s ‘Immortal Game’ against Lionel Kieseritzky,
from 1851.
Anderssen’s 18th move of Bd6 was 87 years later demonstrated by the Moscow
champion Sergey Belavenets to be inferior to the more conservative but decisive Re1
(Damsky, 2002); see Appendix C, Figure C.3. Nevertheless, Anderssen’s game is still
treasured by the chess community because it features a massive and interesting sacrifice
of material despite not being entirely ‘correct’. This game has also been described as a
‘case of brilliance over precision’ (Burgess et al., 2004). Saidy (1972) went as far as
saying that no mechanized computer will ever play like Anderssen did in this game.
More than three decades later, this is still generally true even though game engines like
‘Junior’ (see subsection 3.5.1) could change this. If the assumption is that some degree
of error probably exists in a game’s beautiful combination (and this is reasonable), it
57
can be argued that its beauty is then only enhanced rather than diminished, if no error is
later found.
‘Originality’, while being a desirable trait in a combination, is difficult to ascertain. As
implied in subsection 3.3.2, it is probably unreasonable to expect that a combination be
entirely original in order to be beautiful. In compositions, it is not a convention per se
but used as a safeguard against plagiarism and to minimize ‘anticipations’ (see
Appendix B). In real games, it is an aesthetic plus point, and not a prerequisite.
Originality is also difficult to quantify and evaluate from a computational standpoint
because no single database can hold all previous compositions to compare against (le
Grand, 1986). Therefore, originality is usually left to the discretion and experience of
the judges. For these reasons, it is not considered a viable aesthetic principle that is
equally applicable to both domains.
From the 21 conventions listed in subsection 3.3.1, only 13 are related in some way to
the aesthetic principles. The other 8 (i.e. conventions 2, 6, 7, 9, 10, 13, 14 and 15) are
not. One reason for this may be because these conventions, individually, are difficult to
associate with beauty in the game. For example, convention 2 which states ‘possess a
solution that is difficult rather than easy’ is counter-intuitive to human beauty
perception of objects in general (and even in chess, see section 2.3), which tends to be
more often associated with simplicity (Berlyne, 1972; Girod, 2007). This, however,
does not mean that convention 2 is wrong. It only means that it was probably not
intended for aesthetic purposes, but rather because compositions with simple solutions
are not as challenging to solvers. Conventions 6, 7, 9 and 10 are essentially just
restrictions in terms of game-playing rules whereas conventions 13, 14 and 15 pertain to
‘correctness’ as explained earlier in this section.
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The 4th general aesthetic principle (i.e. ‘spread out the pieces’) is not related to any of
the brilliancy characteristics because there is little control over this in real games, and it
has little to do with the quality of the moves being played. It is a visual property that
only composers have the luxury of incorporating into their compositions, in addition to
the roles played by the pieces. The 5th aesthetic principle (i.e. ‘create geometric
patterns’) is neither related to beauty characteristics nor composition conventions. It is
even more difficult to evaluate as a visual property than the 4th principle because too
many geometric patterns are possible (what would not constitute a geometric pattern in
the game and how would we decide?). Comparisons between these patterns have little
basis in the inherent metrics and properties of the game (see subsection 3.5.2). The other
general aesthetic principles (i.e. 1, 2, 3 and 6, see Table 3.1) can be linked to both
composition conventions and brilliancy characteristics, in varying degrees. This, may in
part, be due to the more explicit nature and higher number of conventions.
3.4 A Selection of Aesthetic Principles and Themes
Not all of the general aesthetic principles listed in the previous subsection were suitable
for computational purposes. Each would eventually need to be formalized in a way that
is flexible enough to accommodate varying implementations of itself, i.e. having unique
values for different configurations of the same principle. This meant that the original list
of six principles (refer subsection 3.3.3, Table 3.1) had to be refined. The notable
changes are explained first (principles 5 and 6), followed by the others (principles 1
through 4), in order.
The 5th general aesthetic principle, i.e. ‘create geometric patterns’ was excluded because
it is very rare in both domains (Levitt and Friedgood, 2008), and difficult to evaluate.
59
The reason it is so rare is probably because, as a strictly visual appeal, it is cumbersome
to incorporate into the required strategic moves of a winning combination. Even though
‘symmetry’ (which can be classified under geometry) has been shown to guide the
aesthetic judgements of beauty, at least with abstract graphic patterns (Jacobsen et al.,
2006), it is not necessarily a prominent or critical aesthetic feature in chess because the
domains are different (see section 2.8).
The 6th general aesthetic principle, i.e. ‘employ chess themes’ was classified as a
separate group because there are even more themes than there are aesthetic principles.
Therefore, themes will now be referred to separately from aesthetic principles. Some
themes pertain more to compositions than real games (e.g. Grimshaw, Pickaninny)
whereas others are common to both domains (e.g. fork, pin). Ten themes from the latter
category were chosen for aesthetic evaluation (see Table 3.2). The themes were chosen
based on those that are usually taught and well-known to players (Averbakh, 1992;
Silman, 1998; Seirawan, 2005). The point of this selection was to minimize bias
between the two domains when it came to aesthetic evaluation. A theme that usually
occurs in real games likely occurs in compositions as well (see Figure 3.1). An
experiment later confirmed this collection of themes to be applicable to both domains,
i.e. not exclusive to either one (see subsection 6.1.1).
The 1st general aesthetic principle (i.e. violate heuristics successfully) was limited to
heuristics that could be properly identified and assessed in the short term (Pritchard,
2000a); consistent with the scope of this research. Four were chosen (see Table 3.2).
Positional and long-term violations of heuristics, such as ‘avoid doubled pawns’,
‘develop the pieces’ and ‘control the centre’ (Berliner, 1990) were not included because
they are not applicable to the chosen scope of mate-in-3. The 2nd principle (i.e. ‘win
60
economically) was separated into four specific principles, the first three of which (see
Table 3.2) were described by Margulies (1977); see section 2.3. The 3rd general
aesthetic principle (i.e., sacrifice material) was retained, as is. The 4th (i.e. ‘spread out
the pieces’) was also retained but supplied with an alternative reference (i.e. ‘sparsity’)
for brevity and clarity. Table 3.2 shows a summary of the refined list of aesthetic
principles and themes (on the right), compared to the general list (left). Detailed
descriptions of all these principles and themes are provided in chapters 4 and 5.
Table 3.2 Refined Aesthetic Principles and Themes
GENERAL REFINED Aesthetic Principles (6) Aesthetic Principles (7)
1
violate heuristics successfully
1
violate heuristics successfully: • keep your king safe; • capture enemy material; • do not leave your pieces ‘en prise’; • increase piece mobility ratio.
2
win economically
2 use the weakest piece possible 3 use all of the piece’s power 4 win with less material 5 checkmate economically
3 sacrifice material 6 sacrifice material 4 spread out the pieces 7 spread out the pieces (sparsity) 5 create geometric patterns EXCLUDED
Themes (10) 6
employ chess themes
1 fork 2 pin 3 skewer 4 x-ray 5 discovered/double attack 6 zugzwang 7 smothered mate 8 crosscheck 9 promotion 10 switchback
The refined list of seven aesthetic principles and now, ten themes, are essentially more
specific descriptions or instances of the six general aesthetic principles listed in
subsection 3.3.3 (excluding ‘create geometric patterns’). They will be assessed
61
individually and cumulatively, to estimate the aesthetic content of a winning chess
combination.
3.5 A Formula for Cumulative Aesthetic Assessment
A formula for the cumulative aesthetic assessment of a move combination is shown in
equation 3.1; (A = total aesthetic value of a combination, P = score for an aesthetic
principle, T = score for a theme, m = the total number of aesthetic principles in the
combination, n = the total number of themes in the combination). A principle or theme
score can be the sum of more than one instance of itself in the combination.
1 1
m nm nA P T= +∑ ∑ (3.1)
The formula is based on the idea that the overall aesthetic value of a chess combination
should equal the sum of the values of the individual aesthetic principles and themes in
the moves of that combination. It stands to reason that as the number of such principles
and themes in a combination increases, so does its aesthetic value. This formula is not
merely a simple quality function because each principle and theme is discrete, and has
its own formalization and score which can vary significantly (see section 3.6). In effect,
these formalizations are quality functions themselves, based on certain metrics and
properties inherent to the game (see subsection 3.5.2). The following subsection first
reviews how standard evaluation functions are developed; it is placed here, rather than
in chapter 2, for proper context. The metrics and properties used in this research are
explained after that.
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3.5.1 The Development of Standard Evaluation Functions
A chess program typically consists of two distinct components namely, the framework
for playing legal chess and the knowledge that allows the program to play chess well
(Schaeffer, 1984; Hartmann, 1989; Hsu, 2004). The former is relatively straightforward
and includes things like legal move generation and (game) tree-searching. The latter is
informal and relies on chess experience. It is difficult to formalize this knowledge and
therefore one of the main factors which make a chess program unique compared to
another. This knowledge is usually described in the form of an evaluation function that
represents certain (weighted) features about the game that influence move selection.
Examples of features include pawn formations, ‘doubled rooks’ (two rooks on a single
rank or file; usually a strong piece formation in the game) and king safety.
Each feature (e.g. after a move) in a position is detected computationally and its
presence multiplied by a particular weight for a cumulative, overall score. The weights
themselves are typically derived through a comparison with a database of master games
or consultation with expert players because some features are more important than
others. The highest score should ideally correspond to the master’s move or advice,
based on which combination of features is being analyzed. The weights are then
optimized to this end which helps to determine terminal node evaluation and move
ordering in the game tree.
More efficient move ordering speeds up searching, e.g. using the alpha-beta algorithm
and better terminal node evaluation aids selection of the best move (Schaeffer et al.,
2007). One of the problems with this approach is that the weights or classifiers are tuned
to the data set analyzed and do not measure the reliability of the game as a whole
63
(Marsland, 1984; Xia et al., 2006; Fernández and Salmerón, 2008). It is not uncommon
for such evaluation functions and their feature weights to be refined and tuned by hand
over many years, based upon careful observation of their performance. The first few
moves, however, typically rely on ‘opening books’ which are also derived from
databases of expert games (Levene and Bar-Ilan, 2007). For endgames with only a few
pieces left, an endgame tablebase is typically used. It contains the solution for all
positions with those pieces. Some endgame solutions, however, can be obtained more
efficiently using a trained artificial neural network (Samadi et al., 2007).
A somewhat different but interesting approach is to use simple ‘atomic features’ which
allow a computer to generate its own evaluation functions (Buro, 1998; Lüscher, 2004).
For example, ‘king in the centre of the board’ can be such an atomic feature but it is not
sufficient by itself to draw any conclusions. If a player has moved his king to the centre
of the board in the beginning of the game, it is considered highly detrimental. It might,
however, be essential to do so in certain endgames. If a second atomic feature such as
‘the presence of only a few pieces on the board’ is added, combining the two makes for
a more meaningful evaluation.
This combination is known as a ‘configuration’ and usually has a positive influence
over the game of the respective player. Each configuration is associated with a value
that represents its impact. These values are determined using a set of rated chess
positions (e.g. based on a checkmate result) that are in turn approximated by the
evaluation function. While the atomic features are hand coded, configurations and their
values are left to the computer. Even with a few atomic features, thousands of
configurations are possible.
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The most important feature in a chess program is usually material (i.e. captured pieces).
Up until the beginning of this century, a computer would almost never sacrifice material
in favour of other positional considerations, such as a well-positioned knight or passed
pawn. Most human players, especially in a tournament setting, would not either. Even
so, newer chess engines such as ‘Junior’ (currently in its 10th edition), factor in
positional evaluation as significant enough to warrant a sacrifice. This is commonly
known as ‘understanding of compensation’ (Ban, 2004). The best example of this was
the bishop sacrifice by (at the time) Junior 9 on the h2 square in game 5 of the
Kasparov-Deep Junior match in 2003 (see Appendix C, Figure C.4); the word ‘deep’ is
added when the program uses multiple processors. The game ended in a draw, as did the
match.
Sacrifices in chess by computers are usually part of highly calculated combinations to
which there is no defence or part of opening book theory. A typical example of the latter
was in game 6 (see Appendix C, Figure C.5) of the famous and emotional, Kasparov-
Deep Blue match (Pandolfini, 1997; Bloomfield and Vurdubakis, 2008). After the last
game of the match (i.e. game 6), Kasparov was reported to have been very upset. He
accused the Deep Blue team of cheating (as in being aided by human experts) and
wanted a rematch but IBM declined and discontinued their computer program (Jayanti,
2005). It is notable that during game 2, Kasparov was amazed but troubled at how
human-like some of the computer’s moves were. This was later attributed to
adjustments the Deep Blue team made to the machine between matches (Fox, 2007).
However, in the Kasparov-Deep Junior game move, the computer had no opening book
theory (to rely on) or winning combination; its evaluation function simply
‘compensated’ for the piece loss with a long-term positional advantage. ‘Rybka’, also a
65
relatively new and top rated chess engine (Ross, 2007), reputedly uses the concept of
‘material imbalances’ in its evaluation function. Material imbalances refer to the slight
differences in piece values compared to the traditional 1-3-3-5-9 model (see subsection
3.5.2(a)). It takes into account for example, rook pawns - which can only capture in one
direction and are thus worth less than the other pawns - and paired bishops, which have
been shown, statistically, to be synergistically stronger than their individual default
values combined (Sturman, 1996; Kaufmann, 1999).
Most of the actual evaluation functions in commercial chess programs are kept secret by
their programmers because it is a highly competitive domain; there are also commercial
interests at stake. While the strength of the evaluation function is critical to a chess
program, its searching capabilities cannot be neglected. Everything else being equal,
programs that are able to analyze more positions in less time typically perform better. It
is notable that while searching at greater depths often yields tangible benefits (Smet et
al., 2003), it can also suffer from ‘diminishing returns’ (Sadikov and Bratko, 2007).
This is when analyzing say, one ply deeper (using considerably more processing
power), does not really yield better results than otherwise. There are also differences in
performance between hardware and software implementations (Hsu, 2006), but a
discussion on this does not belong here.
In some cases, the traditional approach to creating evaluation functions for games is
avoided through the use of simple game parameter inputs to neural networks, and
evolutionary algorithms (Chellapilla and Fogel, 1999; Fogel, 2002; Hingston, 2007).
However, with certain exceptions (Fogel et al., 2006), this method is generally
inefficient in producing programs capable of playing full length games at the world-
class level (Hauptman and Sipper, 2007). Evaluation functions can also be tuned
66
evolutionarily (Bošković et al., 2006; Nasreddine et al., 2006). Even so, none of the
winning programs of the ‘World Computer Chess Championship’ in recent years is
known to use evolutionary algorithms (Weeks, 2008). Evolutionary algorithms,
however, are not limited to Western chess, and results may be better with other variants
of the game (Ong et al., 2007). In any case, an evolutionary approach to tuning the
weights of an evaluation function would likely still be limited by any shortcomings in
the fundamental design of that function.
In principle, the steps involved in creating an evaluation function for game-playing
purposes include: 1) feature definition and selection, 2) feature weighting, and 3) tuning
through supervised or unsupervised learning (Kendall and Whitwell, 2001; Robertsson,
2002; Gomboc et al., 2005). The same basic approach is even used for games like
‘Amazons’ and Othello (Buro, 1998; Lieberum, 2005). In terms of aesthetics, however,
this approach is not necessarily recommended. While feature definition and selection is
possible, weighting is difficult. This is why the inherent metrics and properties of the
game are used in its place (see the following subsection).
It can be seen as approaching the problem from the opposite direction, i.e. using weights
that are (already) widely or universally accepted. Tuning is also difficult because a
reliable data set to compare against (i.e. aesthetically-rated positions) is unavailable.
The aesthetic assessment of a collection of chess combinations by a group of master
players and composers – beyond the resources of the author - could have been used but
was not for four reasons. First, it would likely be prone to bias and personal taste
(particularly between players and composers, see section 3.2). Second, some masters
might be influenced by their familiarity with such combinations and be hesitant to
67
entertain an aesthetic alternative to the typical solution (Bilalić et al., 2008); thereby not
assessing them properly.
Third, formalizing why they found some combinations beautiful (and not others) would
suffer from a bottleneck of knowledge transfer (Michie, 1986; Richards, 2008), which is
only exacerbated by the subject matter (i.e. aesthetics). Fourth, the aesthetic assessment
of masters would not properly reflect the assessment of average players who form the
majority, and are the most to benefit from the implementation of such an aesthetics
model. Hence the advantage of using aesthetic principles based on chess literature, and
universal metrics and properties of the game.
3.5.2 Metrics and Properties Used in the Aesthetic Assessment
Individual formalization was required of all the seven refined aesthetic principles and
ten themes identified (see Table 3.2). This means describing them in the form of
evaluation functions that can capture or approximate their aesthetic content. An
aesthetic principle or theme can be described using certain ‘game properties’ which in
turn can be based on metrics inherent to the game. Figure 3.2 illustrates this.
Figure 3.2 Layers of an Aesthetic Evaluation Function
Game Properties
Game Metrics
Formalization of Aesthetic Principle or Theme
68
The metrics and properties used (see following subsections) are well-established in
chess literature and among players, so this reduces the need for arbitrary or derived
weights to be associated with them. Even though standard evaluation functions also
make use of these metrics and properties, the features they evaluate are often not of
equal value. Some are indeed better than others. Aesthetically, however, this assumption
is avoided between principles and between themes but not within them. In effect, this
means that an aesthetic principle or theme can potentially be just as beautiful as another
but different configurations within the same principle or theme are not necessarily so. It
is easier to distinguish (and with greater consistency), for example, between two
different manifestations of the fork theme than between a fork and a pin. This is because
the criteria used would have to be different for the latter case.
Some chess problem composers might insist that certain themes are better than others
based on say, their complexity. However, this contention is difficult to differentiate
from personal taste, especially when it has been shown that complexity is often not
considered beautiful (Margulies, 1977; Levitt and Friedgood, 2008). Composition
tournament judges, for example, are seldom known to be in agreement on these things
(Wilson, 1978; Albrecht, 2000), even though there are efforts towards establishing
standards (Wenda, 2007). We should also consider what players (unlike composers)
might have to say about the relative beauty of themes.
The approach taken therefore makes no assumptions about this matter and treats them
(i.e. aesthetic principles and themes) equally, especially in light of the variety of
configurations possible within each. Furthermore, the focus area of this research is
aesthetics as it applies equally to both domains of compositions and real games. The
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following subsections (3.5.2(a) through 3.5.2(c)) explain the game properties used -
indicated in the titles - and the metrics they are based on.
3.5.2(a) Piece Value and Piece Count
Shannon (1950) was perhaps the first to explain how a computer could be programmed
to play chess and his ideas are still used as the basis of most modern computer chess
programs (Levy, 1992; Hsu, 2004). He proposed estimated values of the chess pieces so
that a score for every position could be obtained based on the amount of material
captured. This allowed a computer to make automatic decisions as to which moves were
the most favourable, at least from a material standpoint. Modern chess programs still
rely on material as a major criterion when determining the value of nodes (i.e. chess
positions) in the game tree.
The values Shannon ascribed to the chessmen are: queen (9), rook (5), bishop/knight (3)
and pawn (1). A queen is therefore worth 9 pawns and a rook, 5 pawns. They are based
on the ‘pawn unit’ metric. The king is considered to be of infinite value since its capture
means losing the game. However, for practical programming purposes it is usually
attributed a number greater than all the pieces combined, such as 200. Turing (1953) is
also sometimes credited for computer chess even though his piece values were slightly
different (i.e. Q=10, R=5, B=3.5, N=3, P=1). Shannon’s values are generally more
widely used.
Chess programs typically use large factors of these values (e.g. 100 for pawn, 300 for
bishop) and often alter them depending on the current position because in the course of
a real game, a piece’s value may fluctuate depending on how effective it is in said
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position (Silman, 1993). For example, a position with a ‘bad bishop’ (one blocked by
pawns) would have its value for that bishop altered from 3 to perhaps 2, making it
possible for the computer to consider its exchange for a knight or two pawns, not
unfavourable. The Shannon value system for the chessmen is generally a reliable
measure of which side is winning.
It is also reliable in determining how effective a chess manoeuvre such as an aesthetic
principle or theme is because chess players of all levels are familiar with it. Piece values
usually only need to be altered based on the position when a win is unclear and being
sought after during gameplay, not in post-analysis of the combination done for
aesthetics purposes. For this research, the standard Shannon values based on the pawn
unit were therefore used to evaluate the property of ‘piece value’. The king’s value was
reduced to 10 pawn units to keep it aesthetically in line with the other pieces. An
unusually high value would make evaluation of aesthetic principles and themes
involving the king inconsistent with those that do not.
Given that the queen is valued at 9 pawn units - and losing it is tantamount to losing the
game in most cases - the king was valued the minimum amount higher. The qualitative
effect on aesthetics analysis using a different value for the king is not explored in this
research because the chosen value was considered the least arbitrary and most consistent
with the use of the pawn unit metric in this context. In addition to the chessmen, ‘mating
squares’ or squares onto which occupation by an attacking piece would result in
checkmate, were also considered legitimate threats and valued equivalent to the king. In
cases where a piece occupies a mating square, its value defaults to the (higher) value of
the king.
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The ‘piece count’ property is the quantity or number of pieces on the board and is based
on the ‘piece unit’ metric. Piece count can apply in different contexts such as the
number of pieces involved in a particular task, or the number of pieces left on the board.
It is not related to the ‘piece value’ property or ‘pawn unit’ metric because all the
pieces, regardless of their value, count as just one unit. To summarize, the two game
properties described in this section, i.e. piece value and piece count, are based on the
pawn unit and piece unit metrics, respectively.
3.5.2(b) Distance, Piece Power, Mobility and Piece Field
The ‘pawn unit’ and ‘piece unit’ (see previous subsection) are the first and second
metrics, respectively. The third metric is the squares of the chessboard itself, and is used
as the basis of the four game properties described in this subsection. As a unit, the board
squares are a convenient and reliable means of evaluating the property of ‘distance’
(travelled by a piece). Travelling a greater distance means using more of a piece’s
power and is considered beautiful (Margulies, 1977). There are several ways to measure
this. One way is to count the number of squares between two pieces on any line (i.e.
ranks, files or diagonals). This is ideal for long range pieces but not the knight, which
has an L-shaped movement (see Appendix A, Figure A.5).
A second way is to use the Euclidean distance but this was deemed unsuitable due to
fractional values, inconsistent with the chessboard plane. The chosen measurement was
therefore the Chebyshev or chessboard distance. It is equivalent to the number of moves
a king requires to get from one square to another. A knight move is by default 2 squares.
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The distance, d can be calculated as shown below, assuming the two points on the
chessboard have Cartesian coordinates (x1, y1) and (x2, y2).
( )2 1 2 1max ,d x x y y= − −
Distance alone is not a sufficient measure of the power of a piece being used. In order to
evaluate the power of a piece more accurately, a new set of values was introduced based
on the number of squares a piece could possibly control on an empty board. This
property was aptly termed, ‘piece power’. While ‘mobility’ may be an alternative term
for this, it often refers to a piece’s mobility in a particular position (explained below).
‘Piece power’ instead refers to a piece’s inherent mobility, regardless of the position. It
therefore provides a better basis of comparison between positions and between pieces.
By placing each piece on an empty board, piece power was found to be: king (8), queen
(27), rook (14), bishop (13), knight (8) and pawn (4). The pawn’s power is derived from
the fact that it can capture one square to the left or right and move forward one square or
two for a total of four. Appendix A, section 1.1, shows figures depicting the piece
movements; the ‘piece power’ values were based on them. Generally, the more
powerful pieces (e.g. queen, rook) tend to control more squares even though their
‘power’ is not directly proportional to their Shannon value (Euwe, 1982; James, 2007).
Mobility (the position-related kind), is a property relevant to certain principles and
themes. As explained above, it refers to the number of squares a piece controls or can
move to in a particular position. A piece’s mobility in a game is therefore usually less
than its power. The ‘piece field’ property refers to the squares immediately around a
piece, including the one it is on. Pieces in close proximity to each other would have
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overlapping fields. All the four game properties described in this section use the squares
metric.
3.5.2(c) Summary of Metrics and Properties
Table 3.3 presents a summary of the metrics (i.e. pawn unit, piece unit and board
squares) employed in this research, and the properties evaluated (i.e. piece value, piece
count, distance, piece power, mobility and piece field) to which they form the basis.
Table 3.3 Metrics and Properties Used
Piece Type
Pawn Unit Piece Unit Board Squares Piece Value
Piece Count
Distance
Piece Power
Mobility
Piece Field
‘Mating Square’
10 0 0 0 0 4, 6, 9
King 10 1 1 8 1-8 4, 6, 9 Queen 9 1 1-7 27 1-27 4, 6, 9 Rook 5 1 1-7 14 1-14 4, 6, 9
Bishop 3 1 1-7 13 1-13 4, 6, 9 Knight 3 1 2 8 1-8 4, 6, 9 Pawn 1 1 1-2 4 1-4 6, 9
The distance and mobility properties are shown as a range of possible values. The piece
field depends on the location of the piece on the board. For example, a piece in the
corner of the board would have a field of 4 squares (including the one it is on) whereas
one in the centre would have a field of 9 squares. It is illegal for a pawn to occupy a
corner square. The piece count is the same for all the pieces, regardless of their type.
Cumulatively, this property is inherently limited to the number of pieces in a chess set,
i.e. 32. In most cases, the properties mentioned here are used exactly in the way they
have been described. In other cases, adaptations are more suitable (see subsection 4.5.2
for an example). Clarification is provided where necessary.
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3.5.3 A Note on Benchmarks
In order to derive the relative values of certain aesthetic principles and themes against
other implementations of themselves, the concept of a ‘benchmark’ was introduced. It
applies primarily to themes because their ‘strong’ and ‘weak’ configurations are clearer
compared to aesthetic principles. Similar to how ‘piece power’ can be thought of as a
benchmark for the power of a piece employed in particular move (see subsection
3.5.2(b)), a theme benchmark permits comparisons between different configurations of
itself.
The benchmark of a theme is derived from the ideal configuration of a theme (based on
effectiveness and aesthetic characteristics) within the constraints of the board and initial
piece set. Having a benchmark minimizes bias between different themes since the
theoretical maximum score for an instance of any theme would be 1, i.e. if it equalled
the benchmark (even though this score may be exceeded in rare cases, see following
section). Using this method, themes can have dynamic yet consistent scores. For an
aesthetic principle, the benchmark is usually derived from the most relevant parameter
in its evaluation function. The actual benchmarks used are explained in proper context
and in more detail, in chapters 4 and 5.
3.6 A General Methodology for Developing Aesthetics Formalizations
Each identified aesthetic principle and theme needed to be described in computational
terms using the metrics and properties explained in subsection 3.5.2. There is essentially
no mechanical way of doing this. Even chess-playing evaluation functions are often
based on the creativity of the programmer or researcher. However, a general outline of
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the approach can be articulated so evaluation functions for other aesthetic principles and
themes (where it tends to apply more easily, see chapter 5) may be developed in a
similar fashion. The general methodology is as follows:
1. Comprehension of the aesthetic principle or theme.
2. Identification of the pieces that could be involved.
3. Identification of the game properties that apply.
4. Identification of possible liabilities to the principle or theme.
5. Determination of a configuration that reflects the ideal of said
principle or theme.
6. Formulation of an evaluation function so that the ‘strengths’
score for (5) is maximized and its liabilities minimized. The
typical form is shown in equation 3.2. The benchmark is set
to be equivalent to the maximum numerator value so the
score is 1.
( )strengths-liabilitiesbenchmark
score = (3.2)
The 1st step is particularly important. The researcher needs to know what it means in the
context of the game. Experienced players will probably already have a good idea but it
is still recommended that one refer to the relevant chess literature to allay any doubts.
The 2nd step means identifying the types of pieces that could be involved in the principle
or theme. Certain themes such as the pin for instance (see section 5.2), cannot be
executed by short-range pieces like the pawn, knight and king. ‘Mating squares’ may
also be possible threats. A mating square cannot be pinned but it can be forked (see
section 5.1). This step therefore focuses only on the relevant pieces, which later protects
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against unexpectedly trying to factor into the formalization, configurations that could
never happen. It also speeds up computation by skipping over irrelevant pieces.
The 3rd step is an identification of the properties (see subsection 3.5.2) that apply to that
particular principle or theme. The properties should be taken in context of their
relationship with the pieces involved (step 2). For example, ‘distance’ is not a relevant
factor if the pieces involved are short-range. There should be no tendency to look for a
specific number of properties to include. A particular aesthetic principle might require
three different properties whereas another might require just one. Identifying just the
relevant properties of a principle will likely capture more of its aesthetic content in less
time (and using less computational power) than a predetermined set of properties.
The relevant properties are essentially used in the formalization to estimate the
‘strengths’ of a principle or theme from the standpoint of effectiveness and beauty.
Effectiveness is usually in terms of the amount of material threatened whereas beauty
refers to a subset of aesthetic characteristics within the principle or theme itself. For
instance, the distance (relative to its power) at which a piece threatens another is an
illustration of how efficiently it is being used and relates to the concept of economy.
The distance is calculated relative to the piece’s power to differentiate between say, a
queen travelling 7 squares and a bishop doing the same. The latter is considered more
beautiful because less of its inherent power is wasted (Margulies, 1977). Threatening
another piece from a distance is different from the higher level aesthetic principles
related to economy (see Table 3.2) and more to how the concept of economy should
also be taken into account wherever possible within other principles and themes.
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The 4th step is the identification of possible liabilities to the principle or theme. These
include possible manoeuvres by the opponent that may reduce the effectiveness of the
theme. For example, ‘checking’ moves that gain a tempo or intervening pieces that
could come between the attacker and its intended target, reduce the impact of the theme
compared to configurations where there are no such possibilities for the opponent.
Liabilities like this essentially dampen the threat posed by varying degrees, and should
therefore be accounted for in some way.
The 5th step is determining the ideal principle or theme configuration (which can be in
the form of a single position or several positions over many moves, depending on the
principle or theme, and the number of moves required to illustrate it). By default, this
configuration should have no liabilities. Its strengths should be the maximum possible
in terms of the pieces identified in (2) and properties identified in (3). Even so, the
maximum cumulative strength of a theme is sometimes impractical. For example, a
knight can theoretically fork 8 pieces at once but this almost never happens, even in
compositions. In such cases, a reasonably high benchmark needs to be set. To minimize
arbitrariness, it should have some basis in the logic of the theme (see section 5.1). Due
to this, the score for an unusual instance of an aesthetic principle or theme can exceed 1.
This would be justified by its extreme nature.
The 6th step is formulating an evaluation function such that the ideal configuration in
step 5 has a score of approximately 1. This can be achieved by setting the benchmark
(see equation 3.2) as equivalent to the strengths of this ideal configuration (which has
no liabilities or liabilities minimized). It is against this benchmark (also described as a
‘constant’) that all other configurations of the theme will be compared.
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It is difficult and counterintuitive to get human validation of this ideal configuration (as
being the ideal) because often, human players and composers are not sufficiently aware
of what chess literature says about aesthetics. Rather, they rely on their personal taste
and experience. This is the main reason why the approach taken in this research is a
discrete one (i.e. comprised essentially of distinct components) based on relevant parts
of the literature instead of being derived from the polled (general) opinions of a
selection of players and composers. The formalizations presented in the next two
chapters typically follow this general methodology, wherever possible. They are
explained in some detail so the reader can get a better idea how to design evaluation
functions for other aesthetic principles and themes (and construct ideal configurations to
benchmark them against). The following section explains the chosen scope of analysis
in the research and the reasoning behind it.
3.7 The Scope of Analysis Explained
For this research, evaluation of aesthetics was restricted to the analysis of orthodox
mate-in-3 combinations. This was so chosen in order to make meaningful comparisons
between the domains of real games and compositions. Compositions typically present
the solver with a stipulation (task) such as ‘White to play and win’ (e.g. in endgame
studies) or ‘White to mate in 2 moves’ (in direct-mates).
The stipulation for compositions in the scope is a clear task which means that White has
to checkmate the opponent in 3 moves against any defence. Compositions are therefore
of the ‘forced’ or direct-mate variety. For real game combinations, the checkmate need
not necessarily be forced (see subsection 3.3.3). This scope minimizes ambiguity for
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computational purposes because checkmate is an unequivocal and decisive game state;
it is also the ultimate achievement in the game.
Analysis was restricted to one (i.e. the main) variation in the solution. Some
compositions have many variations and feature the unity between them in illustrating a
particular theme (see subsection 3.3.1). However, real games often do not because they
are not ‘planned’ in this way. Comparisons between both domains would therefore be
difficult and perhaps biased, if variations were taken into account. Even so, this does not
mean that adaptations of this model cannot be applied to all the variations of a
combination, if desired.
Shorter move lengths such as ‘mate-in-1’ and ‘mate-in-2’ were not used because they
are limited in terms of the unique ideas and themes that can be demonstrated, even
though it is estimated that there are about 7 billion combinations possible with the latter
alone (Wilson, 1978); though this was unable to be confirmed without any method of
calculation presented. There is more variety in the chosen scope, and compositions of
that type are also widely available. Longer combinations are not as easily available and
furthermore, typically more complex; making them less beautiful to observers
(Margulies, 1977; Levitt and Friedgood, 2008).
Aesthetic evaluation of single moves without a clear objective (e.g. checkmate) is
difficult to gauge, and even more so to compare meaningfully against single moves in
other positions. They are also limited in terms of what can be demonstrated.
Combinations, however, often involve several pieces working together and a noticeable
change in the dynamics of the position (before and after). Combinations without a clear
objective can theoretically be analyzed using some of the aesthetic principles and
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themes selected, but are less reliable as test data because aesthetics is usually taken in
the context of a decisive advantage. Given these issues, the chosen scope was deemed
the most suitable for this research.
3.8 Points of Evaluation (POE)
Mate-in-3 combinations have 3 moves or 5 plies, to be exact. Since White always
checkmates Black (for illustrative purposes), there is no reply to his last move by the
opponent; hence the 5 plies. For this research, aesthetic evaluation was performed at 5
different points (0, 1, 2, 3 and 4) in the move sequence as shown in Table 3.4. These
points are not directly related to the plies (which may also be used for further
clarification).
Table 3.4 Points of Evaluation in a Combination
Points of Evaluation Descriptions
0 Before any move is made (i.e. the initial position) 1 Immediately after the first (key) move by White 2 Immediately after the second move by White
3 Immediately after the third move by White. Applies to everything that directly involves the last piece that moved.
4 The final position. Applies to a broader evaluation of the checkmate and possibly earlier moves.
This categorization is designed to cater for different aspects of analysis involving
aesthetic principles and themes. It also clarifies at which points in the combination a
particular type of evaluation takes place. Some aesthetic principles and themes that
require more than one move to demonstrate are also described as being evaluated in the
final position (i.e. 4). This means that the earlier moves are analyzed after they have all
been played out instead of as they are being played. In rare cases, such as with the
cross-check theme (see sections 5.8 and 5.11), a POE may be described with a ‘.5’ after
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the number (e.g. 1.5 indicating evaluation was done after Black’s reply to White’s
move).
3.8.1 The Moving Piece
In orthodox direct-mate combinations (as opposed to selfmates or helpmates), Black is
obliged to put up the best defence possible and this usually works against the aesthetic
effect achieved by the opponent. Therefore, the bulk of aesthetic evaluation is
performed on the moves made by White and with emphasis on the piece that actually
moved (Wilson, 1978). It is notable that a ‘mirror image’ of any position with White to
play, can also be created with Black to play; so the evaluations are not limited in any
real sense. The manoeuvre by the moving piece must also be ‘favourable’ to be worthy
of assessment, which generally means the following.
1. The piece cannot be captured with impunity by the opponent.
2. The piece is not pinned against its king (which makes it illegal to
move it off the pinning line or in some cases, at all).
3. The principle or theme itself is not compromised (e.g. results in the
overall loss of material).
Items 1 and 3 should be determined to an analysis depth of two plies after the move
played. It is important not to analyze possible variations (e.g. exchanges) too deeply for
mate-in-3 combinations because those from real games are not necessarily forced mates
and do not need to be (see subsection 3.3.3). Such analysis would therefore be pointless
and computationally wasteful.
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3.9 Chapter Summary
A conceptual framework for aesthetics was introduced to enable proper investigation of
the concept in the game of chess. An examination of aesthetics ensued by identifying,
according to the relevant chess literature, what is considered beautiful in both
compositions and real games. These were contrasted against the principles of aesthetics
as also described in the literature. A selection of seven aesthetic principles and ten
themes was made based on suitability and amenability to computation. A formula for
cumulative aesthetic assessment was also presented.
The metrics and game properties used in the research were described, and a general
methodology for developing aesthetics formalizations explained. The scope of analysis
(i.e. mate-in-3 combinations) was elaborated upon, and related issues discussed. Finally,
the ‘points of evaluation’ in a move combination – where aesthetic assessment would
take place – was presented, along with constraints pertaining to the ‘moving piece’. The
following two chapters detail the individual formalizations developed for the seven
aesthetic principles and ten themes, respectively.
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CHAPTER 4: METHODOLOGY – Aesthetic Principle Formalizations
4.0 Formalizing the Seven Aesthetic Principles
In this chapter, the actual formalizations (i.e. evaluation functions) developed for the
aesthetic principles selected for this research are presented in detail. The seven aesthetic
principles are as follows.
1. Violate heuristics successfully.
2. Use the weakest piece possible.
3. Use all of the piece’s power
4. Win with less material
5. Checkmate economically.
6. Sacrifice material.
7. Spread out the pieces (sparsity).
The formalizations are used to estimate the aesthetic value of these principles. As far as
possible, the general methodology of development was used (see section 3.6), even
though it is more applicable to theme formalizations (see chapter 5). The ‘benchmark’
for an aesthetic principle was usually derived from the most relevant parameter in that
principle in a way that provided some scale between 0 and 1 for the overall score. For
example, a piece sacrifice would be compared against the highest value that could be
sacrificed in a single move. Not all the principles, however, have a benchmark per se. In
some cases, a benchmark was not considered the best approach due to the nature of the
principle or how its constituents are related (e.g. ‘sparsity’, see subsection 4.7.3).
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In cases where the formalization is more complex and not as straightforward,
experiments were conducted for validation. These include the principles of ‘checkmate
economically’ (see section 4.5) and sparsity (see section 4.7). ‘Random selection’ of the
real games and compositions used in these experiments was performed based on the
typically sequential but non-overlapping database search results of the relevant criteria
(e.g. Elo rating ≥ 2000, mate -in-3 and 12 pieces); none were hand-picked. In principle,
aesthetic evaluation functions should not be too sophisticated because observers are
probably aware primarily of what they can immediately perceive or remember on the
board (e.g. in terms of depth, area) and would therefore likely base their aesthetic
judgements on that. If the observers are players themselves – which is not unlikely –
their perception and memory might also be influenced by other cognitive factors
(Reingold and Charness, 2005; Ferrari et al., 2007; Linhares and Brum, 2007). These,
however, are beyond the scope of this research and are difficult to account for
computationally.
Evaluation functions for some of these principles (e.g. heuristic violation) possibly
already exist in some form in commercial chess programs but they are used for game-
playing purposes and generally kept confidential (see subsection 3.5.1). In any case, to
suit the aesthetic requirements of this research, all the functions proposed – in this
chapter and the next – were specifically designed by the author based on the relevant
chess literature and game rules (details are in the following sections). Blocks of
pseudocode, showing programmatic implementation of some of these functions, are
provided in Appendix E, for reference. For the equations; v() denotes the value of a
piece, r() its power, d() the distance traversed by a piece in a move or between two
piece locations, and l() the number of legal moves for a piece or player. This
information is presented here to avoid repetition.
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4.1 Violate Heuristics Successfully
Heuristics in chess are general rules that govern good play. A move that violates one or
more heuristics is considered paradoxical (and thus aesthetic) if it results in an
achievement of some kind (e.g. checkmate). Given the scope of analysis (see section
3.7), four heuristics were selected for evaluation (Pritchard, 2000a).
a. Keep your king safe.
b. Capture enemy material.
c. Do not leave your own pieces ‘en prise’.
d. Increase mobility of your pieces.
Other heuristics such as ‘control the centre’ and ‘avoid doubled pawns’ were not
included because they are of little consequence in the short term (Berliner, 1990). In
principle, the more common a particular heuristic is, the more likely its violation will be
noticed by observers and be of aesthetic value to them. For the aesthetics model,
heuristic violations were tested only after White’s key move because by convention
(e.g. in chess compositions) it is the most surprising to observers. The following moves
of the combination may exhibit similar characteristics but the effect is diminished as
checkmate becomes apparent, making heuristic violation more acceptable.
The overall score for the principle of successfully violating heuristics, P1 is calculated
as shown in equation 4.1.
11
1. n
c nP h h−= ∑ (4.1)
h = score for individual heuristic violation(s), hc = heuristic constant (i.e. 4)
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The heuristic constant is derived from the number of heuristics tested for detection (see
subsections 4.1.1 through 4.1.4 below), not the number that actually occurred in the key
move. If the latter was used (i.e. resulting in the average score), it would not properly
account for a higher frequency of heuristic violations. For example, two violations that
score 0.5 and 0.7 individually would score an average of 0.6 whereas a single violation
scoring 0.7 would remain unchanged. The former case arguably deserves a higher score
because of its extra heuristic violation. The proposed formalization solves this problem
with lower but better proportioned evaluations for each (i.e. 0.3 and 0.175,
respectively). The individual heuristics evaluated in this principle are presented next.
4.1.1 Keep Your King Safe
A violation of this heuristic was defined as moving the king to a square which makes it
prone to check on the next move. While not all checks lead to checkmate, players
generally avoid being put in check because ordinarily, a tempo would be lost. Figure 4.1
illustrates the possible areas of the board the king might move to and their
corresponding scores.
Figure 4.1 Scores for Violation of ‘Keep Your King Safe’
0.25 0.5 0.75
1
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The centre four squares are considered a full violation worth 1 point because there is
greater risk of exposure as the king approaches the centre of the board. The surrounding
squares decrease gradually by 0.25 points toward the edges of the board. The score for
this heuristic violation, h1 is as shown in equation 4.1(a).
{ }1 0.25,0.5,0.75,1h ∈ (4.1(a))
Exposure of the king on account of moving a different piece (e.g. to open a line of
attack) was considered too subtle to be a violation of this kind. On an open board in an
imbalanced game i.e. where the opponent has (significantly) more material at some
point of reference, most moves by the king would, in fact, make it prone to check. Some
might even be useful (e.g. to support pawn advancement) but a tempo is still at stake.
There are also cases where the king could have just moved out of check but is still prone
to further check because there was no move that offered protection. Even so, it is not
essential that these possible exceptions be addressed given the scope of mate-in-3
because they would inherently delay checkmate (by White) and are therefore less likely
in such combinations. Programmatic detection of this heuristic violation is
straightforward.
4.1.2 Capture Enemy Material
Not capturing an enemy piece that is exposed counts as a violation on condition that
such a piece could have been captured favourably. This means exchanging a weaker
piece (e.g. pawn) for a stronger one (e.g. knight) or capturing a piece that is undefended.
Defended pieces of equal value do not qualify. A non-capturing move or one that
prefers a piece other than the most valuable available violates this heuristic.
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Aesthetically, pawns are generally not considered pieces worth capturing. They are
often not sufficiently valuable to get sidetracked for, and fall short of what is required
for a decisive advantage (at least between grandmaster games) in chess, i.e. 1.5 pawn
units (Hartmann, 1989). The score for this violation, h2 is calculated as shown in
equation 4.1(b). A full point is scored in cases where a queen’s worth of material was
not captured in favour of some other move.
( ) ( )12 . eh v Q v p−= ∑ (4.1(b))
pe = enemy material not captured, Q = queen
In effect, performing a non-capturing move when several enemy pieces are available
makes the cumulative value of all those pieces a liability. This differentiates between
positions where many opponent pieces could have been captured with impunity and
those where only one or a few could. Capturing the most valuable among them
exonerates the player because only one piece can be captured per move and
heuristically, it is the correct one. Capturing a piece of lower value makes only the most
valuable one a liability because it is heuristically incorrect. The value of the queen was
chosen as the ‘benchmark’ because it is the highest that can be captured in a single
move. Therefore, not capturing the queen is considered the ‘worst’ violation and scores
highest. In certain situations, there may be pieces not captured worth cumulatively more
than the queen, thereby resulting in a score higher than 1.
There is a curious issue, however. In some positions, the white piece in question might
have threatened a black piece that was technically ‘undefended’ because its defender
was pinned (against the king). However, the white piece itself was the one actually
doing the pinning. Therefore, after the hypothetical capture, said defender could legally
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recapture (because it is no longer pinned), making the originally threatened piece
possibly unfavourable and not part of this heuristic violation. The pseudocode for this
function is shown in Appendix E.
4.1.3 Do Not Leave Your Own Pieces ‘En prise’
Leaving your own pieces in a position to be captured (i.e. ‘en prise’) is a heuristic
violation. Like the previous one, it applies only to pieces and not pawns. There is no
violation if the move played captures an enemy piece worth more than the one left ‘en
prise’ or if the friendly piece is favourably defended (i.e. no potential loss of material).
For example, leaving your undefended rook ‘en prise’ to capture the enemy queen
instead (with a different piece) is not considered a violation. This is because material is
still gained after the ‘exchange’. Neither is capturing an enemy knight while leaving
your defended rook prone to capture by his queen because recapturing the queen is
favourable.
There are situations where a player has several pieces left ‘en prise’ and can only save
one. The cumulative value of the others still counts as a liability. More pieces having
been available to the opponent to begin with reflects the paradoxical nature of the initial
position itself (e.g. in a composition) or moves prior (e.g. in a real game). In short, this
violation looks at how much material was neglectfully left at the disposal of the
opponent because heuristically, no piece should be put in that situation. The score, h3 is
calculated as shown in equation 4.1(c). The value of the queen is used as the benchmark
for the same reason as in equation 4.1(b).
90
( ) ( )13 . zh v Q v p−= ∑ (4.1(c))
pz = pieces left ‘en prise’, Q = queen
This violation still applies if the White king happens to be in check (uncommon in
compositions but possible in the starting position of real game combinations). While
that threat takes priority and must be addressed immediately, there are positions where
the king could move out of check and in defence of a piece that is ‘en prise’. The
pseudocode for this function is shown in Appendix E. It is similar to the pseudocode for
‘capture enemy material’ (see previous subsection) and can be combined with it, in
practice.
4.1.4 Increase Mobility of Your Pieces
The last heuristic violation is decreasing your own mobility or permitting the opponent
to increase his. Usually, players try to control more squares with their pieces but
sometimes the opposite is done and this can be quite obvious and puzzling. For
example, a queen or bishop may be moved to the corner of the board behind some
friendly pieces where its mobility is greatly reduced or moved to block several other
pieces, reducing general mobility. The score for violating this heuristic, h4 is calculated
as shown in equation 4.1(d). l(w1) denotes the number of legal moves for White just
before the move played and l(w2) the number of legal moves just after (assuming for the
moment that Black skipped his turn).
( ) ( ) ( )( ) ( )
14 1 1 2
2 1
. ,h l w l w l w
l w l w
− = − <
(4.1(d))
w1/2 = White before/after the move played
91
The violation only applies if there is a reduction in the number of possible moves (to
keep it paradoxical). The worst violation would be leaving oneself with no moves. In
lost positions, this is actually a good tactic to force a draw (stalemate) if the opponent
fails to notice but in forced mates, it is unusual.
4.2 Use the Weakest Piece Possible
Using the weakest piece possible to achieve a particular objective is considered
aesthetic from an economic standpoint. Given the scope of analysis, this was limited to
checkmate as performed on the last move of the combination. Only with a clear
objective like checkmate can a computer determine conclusively what a piece is actually
being used for. This is why evaluation of this principle at other points in the
combination was avoided. The formalization for this principle, P2 is as shown in
equation 4.2.
( ) 12 .c cP wp r p −= (4.2)
pc = checkmating piece, wpc = principle constant (i.e. 4)
The principle constant is set to 4 so if a pawn turns out to be the checkmating piece, the
score reaches its maximum of 1. In the case of a double checkmate i.e. two pieces
attacking the king simultaneously with mate, only the piece that moved i.e. the critical
piece, counts (Margulies, 1977). For a ‘two-way discovered checkmate’ (see Appendix
C, Figure C.6), the weaker of the two pieces is chosen. This principle applies to all
pieces except the king because it cannot be used to checkmate.
92
4.3 Use All of the Piece’s Power
Using all of the piece’s power can be interpreted as the number of squares a piece
traverses in a single move. Travelling a greater distance is considered more beautiful
than a shorter one (Margulies, 1977). If a less powerful piece (e.g. bishop) travels a
certain distance, more of its total power is used than if a more powerful piece (e.g.
queen) travels the same distance. Therefore the bishop move is considered more
beautiful than the queen move. This principle applies to all of White’s moves in the
move sequence and a broader evaluation of the final position, i.e. n=4 (see section 3.8).
The score for this principle, P3 for all the points of evaluation, n in the combination, is
calculated as shown in equation 4.3.
( )( ) ( )( )( ) [ ] ( ) ( )
13
1. ,
4 ( ) ( , )
nn n
k
P d mp r mp
if n then d mp d cp b r mp r cp
−=
= = ∧ =
∑ (4.3)
mp = the moving piece, n = evaluation point
cp = the checkmating piece, bk = the black king
The final position applies to the distance between the checkmating piece and the enemy
king. This was done because checkmates are sometimes performed from afar and these
are considered more beautiful. The knight - given its unique movement - defaults to a
fixed Chebyshev distance of 2. The piece that moved at n=3 is not necessarily the one
that delivers checkmate e.g. a ‘discovered checkmate’ (see section 5.5). In the case of a
‘two-way discovered checkmate’ (see Appendix C, Figure C.6) or a double checkmate
(see section 5.5) the first of the two pieces (upper left to lower right of the board, as in
FEN) is chosen. It is possible in certain positions for the total score of this principle to
exceed 1 (e.g. two maximal pawn moves + one knight move + mate using knight = 1.5)
or fall significantly below it (e.g. two single square queen moves + one single square
93
rook move + mate using rook right next to the king = 0.22). Most positions, however,
tend to score between 0 and 1. It is notable that all positions, regardless of their beauty,
will have a score for this principle.
The mean power of the pieces in the combination was not used as the score because this
principle is evaluated in all of the moves. First, the aesthetic effect of a long-range
move (e.g. across more than half the board) - usually not more than a single one in
practical checkmate combinations - would otherwise be unnecessarily diminished by the
other short-range ones, which are also evaluated. An arbitrary definition of what
constitutes a ‘long-range’ or ‘short-range’ move was avoided.
Second are the limitations of the board itself. It can be argued that a bishop or queen can
at most traverse a distance of 7 squares so its benchmark (i.e. its piece power) should
not exceed that amount. However, this then leaves no distinction between the
movements of the queen, rook and bishop. Given the scope of analysis (see section 3.7),
the total score of a combination for this principle is usually between 0 and 1, i.e. in line
with the other principles. For longer combinations, the recommended option would be
to evaluate only 2 or 3 of the longest-range moves. These are likely to exhibit the
aesthetic principle most prominently and would ensure the score remains consistent (i.e.
between 0 and 1). The pseudocode for this function is shown in Appendix E.
4.4 Win with less Material
This principle is considered aesthetic because it is paradoxical (Margulies, 1977; Levitt
and Friedgood, 2008). Usually, the side with more material is more likely to win. It
applies only if the total material value for Black exceeds that of White because the
94
winning side should not be penalized aesthetically for not exhibiting this principle. It is
evaluated only in the initial position before any moves are made. The kings are not
included because their presence on the board is mandatory. The score for this principle,
P4 is calculated as follows (equation 4.4).
( ) ( ) ( ) ( )14 . ,c b w b wP wl v p v p v p v p− = − > (4.4)
wlc = principle constant (i.e. 38), pb/w = all Black/White pieces in the position
The principle constant or benchmark is set to 38 because this is the maximum amount of
expendable material for an army (at least one pawn must be left) where checkmate is
still possible, however unlikely. Theoretically, Black would have some material on the
board that would blockade his own king and facilitate the checkmate; though this does
not take into account the possible promotion of that lone white pawn later in the
combination. The possibility of such a mating combination (i.e. with a score of 1 for this
principle) is interesting in its own right. Setting a lower benchmark might appear more
reasonable in giving this principle a chance of achieving scores comparable to some of
the other principles and themes. However, it is difficult to determine what the value
should be so the theoretical constraint within the game itself (i.e. 38) was used. It is not
so unreasonable once one takes into account the possibility of initial positions where
Black already has more than one queen (e.g. from pawn promotion).
4.5 Checkmate Economically
The aesthetic principle of economy, in general, is relatively more complex than the
previous four principles. The following subsections address the relevant issues.
95
4.5.1 Explanation of the Concept
Economy implies using no more resources than necessary to achieve a particular
objective. In chess, the objective is simply to checkmate the enemy king and the
resources for this are the chessmen. Economy in chess, however, can have a few
meanings (Rice, 1997). The first and most common is ‘economy of material’ which
means not using more pieces than necessary to achieve checkmate. The second meaning
is ‘economy of space’ which is using the chessboard to its fullest as opposed to
cramming all the pieces into one corner. This can also be interpreted as a preference for
sparse positions over crowded ones and is addressed in section 4.7. The last is ‘economy
of motivation’ which is keeping all lines (i.e. variations) in the solution (e.g. of a chess
problem) relevant to the theme. This interpretation is rather limited to compositions
since multiple variations and employing themes are primarily composition conventions.
Usually, economy in chess assumes its default meaning (i.e. material) and this is the
definition used here (Hooper and Whyld, 1996). Even though economy may be evident
in the moves preceding checkmate, the final (mated) position best encapsulates the
concept (Nunn, 2002). It is in this position therefore that economy was evaluated. It is
difficult to ascertain economy in the moves preceding the final position. This is because
they may contain piece sacrifices or manoeuvres that - unless the intention is known -
could be construed as exemplifying other aesthetic principles. It is also difficult to tie
the ‘goal’ in those moves to the checkmate itself in a meaningful way.
Economy of material generally refers to the number of pieces used in the checkmate but
some references go a step further to consider also the amount of material these pieces
add up to and their relationship with one another (Howard, 1967; Levitt and Friedgood,
96
2008). For example, using a queen to do a pawn’s job (e.g. blocking the advance of an
enemy pawn) would be unsound economically because not only is the queen considered
more powerful in most positions but also that it should have had more responsibilities
for that reason. Such positions are not uncommon in chess (Tabibi and Netanyahu,
2004). In some positions, this imbalance might be necessary for checkmate but in a
strict sense that does not improve its economy. Several pieces working together in a
combination can be seen as economical even though their cumulative material value
might be higher than that of the queen.
4.5.2 Features of Economy
Certain economic features or properties were identified for the development of the
economy evaluation function. The first is the ‘piece count’ used to achieve checkmate.
If more pieces are involved, the less economical a position is considered to be.
‘Involved’ here means participating directly in the checkmate. Removing a piece that is
involved would therefore invalidate the mate. The second feature is the ‘piece values’
(see subsection 3.5.2(a)) and the third, their ‘powers’. Piece ‘power’ here does not refer
to the game property as described in subsection 3.5.2(b) per se. It is an adaptation of it
used in the context of ‘piece field’ (same subsection). This is illustrated in section 4.5.3.
The third feature stems from the conventions employed in Bohemian problems and is
explained as follows. Chess problems can typically be divided into three ‘schools’
namely Bohemian, Logical and Strategic (see subsection 3.3.1). These are basically
different styles of composition.
Unlike the other two, the ‘Bohemian’ school of problems places a strong emphasis on
economy (Howard, 1967; Zirkwitz, 1994). The ‘Logical’ and ‘Strategic’ schools do not
97
neglect economy but offer more flexibility in favour of themes that might otherwise be
constricted. Composers of the ‘Bohemian’ school are particularly interested in what are
known as ‘model mates’ which typically include the following conditions.
• No square in the mated king’s domain is guarded more than
once or is blocked as well as guarded.
• All of pieces of the winning side (with the possible
exception of king and pawns) participate in the checkmate.
• If an enemy piece is in the king’s domain but is pinned, it is
exempted from the first condition.
The first condition on its own is also the criterion for a ‘pure mate’. The ‘domain’ of a
piece, similar to but not exactly like its ‘field’ (see subsection 3.5.2(b)), includes all the
squares immediately around it, excluding the one it is on. The term is used here to
explain the model mate. In ‘ideal mates’ (not characteristic to any particular school) all
the pieces, including those of the opponent, are used. These conditions illustrate the
need for piece power (in relation to piece fields) to be taken into account. This
essentially means the number of squares in the king’s field (not just domain), that can be
controlled; see following subsection.
4.5.3 The Economy Evaluation Function
The three economic features mentioned in the last subsection were incorporated into the
formalization for the principle of economy, P5 (see equation 4.5). The ‘control field’ of
a particular active piece (an) refers to the number of squares in the enemy king’s field an
‘active piece’ controls or guards i.e. one that is essential to the checkmate and cannot be
98
removed without invalidating it. The term king’s field here refers to its ‘piece field’ as
described in subsection 3.5.2(b). ‘Control field’ is a subset of squares within the king’s
field. Passive pieces are those that can be removed without invalidating the checkmate.
( ) ( )( )1 1 15
1 1. . .n n
n n n kP p a f o s f− − − = − + ∑ ∑ (4.5)
an = control field of a particular active piece, fn = maximum control field of that active
piece, o = number of overlapping control field squares, fk = standard king’s field,
sn = maximum control field of a particular passive piece,
p = number of friendly pieces on the board
The maximum control field of a particular active piece (fn) is equivalent to the number
of squares in the king’s field it could at most control. These values were determined by
testing each piece on the board and placing it in proximity to the enemy king. It was
found to be: king (3), queen (6), rook (4), bishop (3), knight (2) and pawn (2). Figure
4.2 shows how they were determined. Nowhere else on the board can each individual
piece control more squares in the king’s field.
(a) King (b) Queen (c) Rook
(d) Bishop (e) Knight (f) Pawn
Figure 4.2 Maximum ‘Control Fields’ for the Chessmen
99
The square onto which the piece in question itself resides is not considered part of its
‘control field’ because it is not defended. In the case of Figure 4.2(b) and 4.2(c), the
queen and rook would ordinarily have to be defended by a friendly piece to prevent
their capture by the enemy king. Long-range pieces (i.e. queen, rook and bishop) are the
only pieces that can control a square in the king’s domain through the king. This is
because the king would still be in check if it moved to the square immediately behind it
(i.e. in the line of attack). The main parameter of the evaluation function therefore is the
summation of the active piece control fields against their respective maximum control
fields. The closer a piece is to its maximum control field capability, the less of its power
is wasted and its application in the mate deemed more economical.
Secondary parameters include overlapping control field squares (o) and the presence of
passive pieces on the board. The overlapping control field squares are those in the
king’s field which are empty yet guarded by more than a single white piece. The
maximum control field of a passive piece (sn) is the number of squares it could have
controlled. Even though a king on the edge of the board has a field (fk) of 6 squares and
one in the corner only 4, the field for a king in the centre of the board (9 squares) was
chosen. This is because it is considered more beautiful and better in terms of conformity
to the first condition of the model mate (le Grand, 1986); see previous subsection.
After the secondary parameters have been subtracted from the primary one, the result is
divided by the total number of friendly pieces (p) on the board. Even though the king
and pawns are permissible exceptions in model mates, for consistency the evaluation
function does not exclude them. The main reason for the permissibility afforded to kings
and pawns in the Bohemian school of composition (at the expense of economic purity)
is that composers felt rather restricted otherwise. It is difficult, perhaps impossible, to
100
compose novel problems featuring interesting themes if every piece must strive for
economy.
The evaluation function therefore deals with economy (of the checkmate position) in its
purest sense. While the evaluation function typically returns a positive value between 0
and 1 for most checkmate positions, there are situations in which it returns a negative
score. One option was to normalize all negative scores to zero, implying that an
economic score less than zero would simply mean ‘poor economy’ but this meant being
unable to compare poor positions with each other so negative values were retained. In
effect, such positions can have economy scores that subtract a little from its overall
aesthetic evaluation, unlike the preceding aesthetic principles.
4.5.4 The Process of Evaluation
The process of evaluating the economy of a checkmate position requires seven steps.
They are summarized as follows.
i. Determine and remove passive pieces from the board.
ii. Sum the ratio of the remaining active piece control fields
against their maximum fields.
iii. Count the number of overlapping control field squares.
iv. Sum the number of theoretical maximum control field
squares for the passive pieces.
v. Add (iii) and (iv) and divide by the standard king’s field.
vi. Subtract (v) from (ii).
vii. Divide (vi) by the number of friendly pieces on the board.
101
Step (i) is accomplished by systematic removal of pieces from the chessboard starting
from the most valuable (queen) and continuing in descending order, based on their
value. If duplicate pieces such as two rooks or four pawns exist, the order of removal is
from the upper left of the chessboard (coordinate a8) to the lower right (h1). This is
similar to the sequence employed in the standard Forsyth-Edwards Notation (FEN) used
in the game (see Appendix A, subsection 1.4.1). The same process is then repeated
starting with the weakest piece. This two-phase approach is necessary to reduce mate
configurations to their truly essential pieces. It was found that in approximately 1% of
checkmate positions, a single phase approach is insufficient. For example, in the
position (FEN: K3R3/1Q1N4/3pPp2/2pk2n1/2pbnp2/4pbr1/3p1r2/8 b) taken from the
#3 composition (Gerald Frank Anderson, Western Morning New, 1922), the pawn’s
removal last (in the first phase) only afterwards renders the knight passive (see
Appendix C, Figure C.7). A second phase starting with the strongest piece again would
theoretically also work. The pseudocode for this function is shown in Appendix E.
Figure 4.3 shows the economy evaluations of two checkmate positions. One is taken
from a real game and the other from a composition. In the real game (Figure 4.3(a)),
there are many passive pieces namely the rook on d7 and all the pawns except for the
one on g3. This means they can be removed without invalidating the checkmate. The
king does not play a role in the mate either and represents an additional economic
liability. In the composition (Figure 4.3(b)), all of the pieces are active - none can be
removed without invalidating the mate - and it is only a question of how much of their
power is being used to that end. Chess problems are usually composed with some sense
of economy in mind whereas real games have little control over it.
102
XABCDEFGHY
8-+q+-+-+(
7+-+R+ptr-'
6l+-tR-+-+&
5zp-+Qmk-zp-%
4Pzp-+-+-+$
3+-+-+-zP-#
2-zP-+-zP-+"
1+-+-+-mK-!
xabcdefghy
XABCDEFGHY
8k+-+-+-+(
7+-+-+-+-'
6NmK-+-+-+&
5+-+L+-+-%
4-sN-+-+-+$
3+-+-+-+-#
2-+-+-+r+"
1+q+-+-+-!
xabcdefghy
(a) Obodchuk vs Konnov RUS-ch sf, Orsk, 2000 Economy Score: 0.128
(b) C. S. Kipping Manchester City News, 1911
Economy Score: 0.375
Figure 4.3 Economy Scores of Checkmate Positions
4.5.5 Validation
Given that the evaluation function for economy is more complex than those for the other
aesthetic principles and more difficult to ascertain, two experiments were performed to
ensure that the principle was being reasonably captured.
4.5.5(a) Compositions vs. Tournament Games
The first experiment compared one thousand randomly selected checkmate positions
from generic compositions (i.e. not belonging to any one school of composition) against
the same number of randomly selected tournament game checkmate positions (Meson
Chess Problem Database, 2008; Mega Database, 2008). Compositions are considered
more economical than real games because of the requirements imposed by convention
(see subsection 3.3.1). They also have the benefit of a composer who is furthermore not
under official time constraints (like games in a chess tournament). The experiment
103
therefore intended to see if compositions scored higher economically, than tournament
games.
For the compositions, where more than one checkmate variation existed, only the main
line was chosen. In the few cases where the main line was unknown, one was selected at
random. For the tournament games, only those between expert players (Elo rating ≥
2000) were used to minimize bias. Experts typically have a rating of 2000 and above. A
minimum rating of 2300 usually qualifies for an FIDE Master title whereas
grandmasters typically have a rating of 2500 and above (Kasparov, 2007). Amateur
games would likely feature less economical checkmates due to unsound play. Highly
rated players are more likely than less experienced ones to make better use of their
pieces. The result of the experiment is shown in Figure 4.4.
Figure 4.4 Economy Scores for Compositions and Tournament Games
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 51 101 151 201 251 301 351 401 451 501 551 601 651 701 751 801 851 901 951
Scor
e
Position
COMP
TG
104
The scores have been sorted in ascending order for clarity. The compositions scored a
mean economic value of 0.299 (SD 0.2) compared to the tournament games which
averaged only 0.089 (SD 0.18). The difference of 0.21 in means was statistically
significant (TTUV, 2T, SL 1%); t(1987) = 24.89, P<0.01. For the statistical
abbreviations used, see Abbreviations; t(degrees of freedom) = t-stat, p-value. This
difference in the means suggests that the evaluation function is able to discern correctly
between the economic values of typical checkmate positions in compositions and
regular games (Ruxton, 2006). However, this could be attributed mainly to the
preponderance of passive pieces found in regular games. This brings us to the second
experiment.
4.5.5(b) Compositions vs. Tournament Games (Improved)
A second experiment was conducted but this time with positions from both groups
‘improved’ (i.e. passive pieces removed from the board). These would better contrast
the finer aspects of economy between the two groups, if any (see Figure 4.5).
Compositions in this case scored an improved mean of 0.546 (SD 0.14) and tournament
games a much larger improvement to 0.448 (SD 0.15). The difference of 0.098 was still
statistically significant (TTUV, 2T, SL 1%); t(1981) = 15.05, P<0.01. The tournament
games had an average increase of 0.359 (SD 0.13) economic points compared to 0.247
(SD 0.18) points for the compositions. Of the tournament game checkmates, 96%
benefited economically from the removal process but only 77.5% of the compositions
did. The result of this experiment concurs with the idea that even without passive
pieces, chess problems (having the benefit of human composers), still exhibit better
overall economy than real games.
105
Figure 4.5 Economy Scores for ‘Improved’ Positions
The difference in means (0.149) between the unimproved compositions (0.299) and
improved tournament game positions (0.448) was also statistically significant (TTUV,
2T, SL 1%); t(1882) = -18.927, P<0.01. This suggests that passive pieces are a severe
economic liability even to compositions, and that without them even real game
checkmates are of comparable or better economic value. However, any real world
application of the evaluation function would have to take tournament games and
compositions as they are (as in the first experiment).
4.5.5(c) Testing against Human Assessment
Computational evaluation of economy was not tested against human evaluation of
economy because there was reason to believe the latter would be inconsistent with chess
literature (see subsection 4.5.1). An informal survey with respondents from an online
chess community suggested this (Gameknot, 2008). Many respondents could not
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 51 101 151 201 251 301 351 401 451 501 551 601 651 701 751 801 851 901 951
Scor
e
Position
COMP
TG
106
understand what the author meant by ‘economical’ and preferred specifics. Doing so,
however, would have compromised the survey. The experiments performed were
therefore considered sufficient to validate the proposed formalization for economy.
It is noteworthy that validating every aesthetic principle and theme (see chapter 5)
against human player assessment would not have been feasible for two reasons. First,
the resources available to the author were limited. Second, such validations would likely
have compromised the necessary tests for correlation between cumulative computational
aesthetic evaluation and human player aesthetic assessment (see section 6.6) by asking
leading or revealing questions about individual principles and themes (Ahmad, 2007).
4.5.6 Minor Economical Differences
Minor economical differences between two positions are difficult to confirm
experimentally. In most cases, they are probably irrelevant but it is preferable that a
formalization of economy be able to account for them. This section illustrates how the
proposed formalization evaluates such differences. Several checkmate positions were
chosen and tested by systematically making minor modifications that are consistent with
chess literature in terms of improving the position economically or otherwise. Figure 4.6
shows an example. In each case, the evaluation function reflected the improvement (or
otherwise) one would expect from the modification.
In Figure 4.6(a), the original tournament game checkmate can be seen as rather
inefficient which is why it scores poorly. In 4.6(b), the passive rooks are removed which
means fewer pieces are on the board so the score improves. In 4.6(c), the unnecessary
pawns are removed and the economic score increases even further. Finally in 4.6(d), the
107
enemy bishop on g7 is removed and the king is made to participate, increasing the score
to 0.57407. The bishop was removed so the king has a more obvious role to play by
guarding the g7 square. Even if the bishop was not removed and the king became
unnecessary to the checkmate, the position would still score better than 4.6(c) (i.e.
0.463). This is because the king simply cannot be removed from the board and in the
strictest sense of economy is better contributing something to the checkmate (e.g.
doubly guarding the g8 flight square) for its presence than sitting idly on g2.
XABCDEFGHY
8-+-wQ-mk-+(
7+-+-+-vl-'
6-+-+-trP+&
5zpp+p+l+-%
4-+-wq-+-+$
3+-+-+-tR-#
2P+-+-+KzP"
1+-+-+R+-!
xabcdefghy
XABCDEFGHY
8-+-wQ-mk-+(
7+-+-+-vl-'
6-+-+-trP+&
5zpp+p+l+-%
4-+-wq-+-+$
3+-+-+-+-#
2P+-+-+KzP"
1+-+-+-+-!
xabcdefghy (a) Westerinen vs. Matanovic
Helsinki zt, 1972 Score: -0.02381
(b) Unnecessary pieces removed
Score: 0.14444 XABCDEFGHY
8-+-wQ-mk-+(
7+-+-+-vl-'
6-+-+-trP+&
5zpp+p+l+-%
4-+-wq-+-+$
3+-+-+-+-#
2-+-+-+K+"
1+-+-+-+-!
xabcdefghy
XABCDEFGHY
8-+-wQ-mk-+(
7+-+-+-+K'
6-+-+-trP+&
5zpp+p+l+-%
4-+-wq-+-+$
3+-+-+-+-#
2-+-+-+-+"
1+-+-+-+-!
xabcdefghy (c) Unnecessary pawns
removed Score: 0.38889
(d) King made to participate
Score: 0.57407
Figure 4.6 Minor Economic Improvements to a Position
108
4.5.7 Paradoxical Economy
There are situations in which economic improvement might not be as obvious as they
are in Figure 4.6. Figure 4.7(a) shows a constructed position where White has material
on the board worth 11 points (queen + 2 pawns) whereas in 4.7(b), the pawn on b5 is
replaced with a bishop on d7, totalling 13 points in material.
XABCDEFGHY
8-+-+-+-+(
7+-+-+-+-'
6-+-+-+-+&
5mKPmk-+-+-%
4-+-wQ-+-+$
3+-+-zP-+-#
2-+-+-+-+"
1+-+-+-+-!
xabcdefghy
XABCDEFGHY
8-+-+-+-+(
7+-+L+-+-'
6-+-+-+-+&
5mK-mk-+-+-%
4-+-wQ-+-+$
3+-+-zP-+-#
2-+-+-+-+"
1+-+-+-+-!
xabcdefghy (a) Material: 11 Score: 0.69444
(b) Material: 13 Score: 0.70833
Figure 4.7 Economy Paradox
On the surface it would appear that 4.7(a) should be more economical than 4.7(b)
because there is less material on the board and the number of pieces remains unchanged.
However, the evaluation function attributes a marginal advantage to 4.7(b) on account
of the ratio of power used by the bishop (2 out of a maximum of 3 squares controlled in
the king’s field). This is more compared to the b5 pawn in 4.7(a) which uses less of its
power (1 out of 2 squares) even though the bishop in 4.7(b) does introduce an extra
overlapping control square on b5 that was not present in 4.7(a).
109
This example illustrates that there is actually more to economy than simply having less
material on the board despite how paradoxical it may seem. Human players might find it
difficult to make this distinction especially when the difference between two such
positions is minor. The economy evaluation function was designed around the concept
of ‘perfect economy’ as a theoretical maximum for any position. Even though it is
uncommon for a position to score less than 0 economically, it is not possible to exceed
1.0. The evaluation function (see equation 4.5) is designed this way.
There are some other things to consider as well with regard to how checkmate positions
are evaluated in terms of economy. Looking at the four main variables of the evaluation
function (i.e. active pieces, overlapping squares, passive pieces and number of friendly
pieces on the board) it can be seen that they are not entirely independent of one another.
Introducing an extra piece to a checkmate position for example, would necessarily
increase either the active pieces or passive pieces on the board which can have a variety
of effects on the score. Especially if it introduces more overlapping control squares or
reduces them (e.g. by blocking another piece). In principle, altering the piece
configuration even slightly can be significant economically.
4.5.8 Perfect Economy
The highest economic value is obtained from positions that use all the power of all the
pieces with no overlapping control fields. Figure 4.8(a) shows the highest economic
value discovered in a Bohemian composition during experimentation and the highest
one that the author managed to construct.
110
XABCDEFGHY
8-+-+-+-+(
7+-+-+-+-'
6-+-+-zpp+&
5+-+-+-+-%
4-+-+k+-+$
3+-+-+Q+-#
2-vL-+K+-+"
1+-+-+-+-!
xabcdefghy
XABCDEFGHY
8ltr-+-+-+(
7zpk+K+-+-'
6-tr-+-+-+&
5+-sN-+-+-%
4-+-+-+-+$
3+-+-+-+-#
2-+-+-+-+"
1+-+-+-+-!
xabcdefghy (a) E. Shulz
Dortmunder Zeitung, 1937 Score: 0.81481
(b) Constructed Position
Score: 1.0
Figure 4.8 Highly Economical Checkmates
Figure 4.8(b) is an example of ‘perfect economy’ for White. The squares c8, c7 and c6
are all controlled by the white king whereas the knight checkmates the enemy king and
prevents its flight to a6. Both pieces are contributing all of their available power to the
checkmate and there are no more friendly pieces on the board than necessary. With the
exception of a pinned piece that would keep active an otherwise passive white piece,
black pieces do not affect the economic score.
4.6 Sacrifice Material
The principle of sacrificing material is paradoxical but not the same as violating the
heuristic of leaving your own pieces ‘en prise’ (see subsection 4.1.3) because it applies
more to exchanging your pieces unfavourably for positional superiority (Sukhin, 2007).
This is usually done to secure a decisive advantage or force a win. The ‘romantic’
players of the late 18th and early 19th century often used bold sacrifices that were not
always sound to impress spectators (Guid and Bratko, 2007).
111
Former world chess champion Mikhail Tal, who considered chess first and foremost an
art, was also known for intuitive sacrifices that gave rise to complications on the board
and confused his opponents (Gallagher, 2001). In 21st century chess, however, sacrifices
are not as popular in real games or compositions because computer analysis can easily
reveal flaws in them. Sacrifices are still employed when necessary but are more
calculated and scrutinized than was done in the past. The impact of a sacrifice (on the
spectator) usually correlates with the amount of material lost. The function below
(equation 4.6) is used to calculate the value for this principle.
( ) ( ) { }( ) ( )
16 .. , 9,14,19.. ,c i f i f c
i f i f
P sm w w b b sm
w w b b
− = − − − ∈ − > −
(4.6)
w/bi/f = the total value of White/Black initial/final material,
smc = principle constant
The principle constant, smc depends on the number of moves there are in the
combination. For example, a mate-in-2 sequence would have a principle constant of just
9 because this is the most amount of material (i.e. a queen) that could be lost to the
opponent in that span of moves. A mate-in-3 would have a material constant of 14 since
after the opponent’s second move, at most another rook (given the original piece set),
could be lost and so forth. No sacrifices are possible for mate-in-1 positions. Given the
scope of analysis, smc = 14.
The function (equation 4.6) takes into account sacrifices of any number of pieces of any
type (by White), yet is also sensitive to changes in material balance due to exchanges
and pawn promotions (by both sides). This is because the difference in material at the
end of the move sequence will properly reflect how much material was actually lost (in
a ‘clean’ sense) by White in the combination. It would be misleading, for example, to
112
sacrifice a knight on the first move but on the second, promote a pawn to a queen. A
sacrifice is therefore determined and evaluated in the final position, after all the moves
have been played out. Negative values indicate that White actually gained material but
like in P4 (i.e. ‘winning with less material’, see section 4.4) he is not penalized for it.
Many mating combinations in fact, necessarily result in significant material gain by the
winner.
4.7 Spread Out the Pieces (Sparsity)
Similar to ‘economy’ (see section 4.5), the principle of ‘sparsity’ is relatively more
complex than the other aesthetic principles. It also has a potentially wider and more
immediate scope of application, i.e. not limited to the game of chess. The following
subsections address the relevant issues.
4.7.1 Explanation of the Concept
Crowded or cluttered positions in games are generally considered less beautiful than
sparse ones because the relationships between the pieces become more complex and
difficult to see (Troyer, 1983; White, 2003). This should not be confused with the
complexity of the position in the usual sense. That can be measured by looking at how
difficult it is (e.g. for a computer) to choose the best move in a position as it analyzes
deeper. A larger cumulative (absolute) difference between the evaluation scores of the
best and second best moves as search depth increases usually indicates greater
complexity of the initial position (Guid and Bratko, 2007).
113
Two important factors when evaluating sparsity are: 1) the number of pieces on the
board, and 2) their proximities to each other. Ideally, a game position can have many
pieces yet be considered just as sparse as a position with only a few, depending on its
configuration. It can also have its pieces spread out far enough apart to be considered
‘sufficiently sparse’ (i.e. no longer cluttered), regardless of any further efforts at
improving it. In Figure 4.9(a), there are 7 pieces on the board. The position would not
be considered crowded because the pieces are relatively far apart.
XABCDEFGHY
8k+-+-+-+(
7+-+-+-+-'
6N+N+-+-+&
5+-+L+-+-%
4K+-+-+-+$
3+-+-+-+-#
2-+-+p+r+"
1+-+-+-+-!
xabcdefghy
XABCDEFGHY
8-+-+-+-+(
7+-+-sN-+-'
6-zp-+-+-+&
5+-+-+-mK-%
4-+nmk-+-+$
3+-+-+-wQ-#
2-sn-+R+-+"
1+-+-+-+L!
xabcdefghy (a) C.S. Kipping,
Manchester City News, 1911 (b) Godfrey Heathcote,
British Chess Magazine, 1906 XABCDEFGHY
8n+ltRN+L+(
7+Pzpp+p+r'
6pzp-+k+N+&
5+r+-zp-zP-%
4n+P+P+Kzp$
3+-+-+-+-#
2-+-+-vl-+"
1+-+-+-+-!
xabcdefghy
XABCDEFGHY
8n+l+N+-tr(
7zp-+-mk-+-'
6-zP-zp-zpN+&
5+-zp-zp-zP-%
4-zp-+-+-+$
3sn-+RzP-mKp#
2-+P+-+-+"
1+r+-vL-vl-!
xabcdefghy (c) K. Fabel,
Deutsche Schachzeitung, 1965 (d) K. Fabel,
Deutsche Schachzeitung, 1965 (Modified)
Figure 4.9 Sparsity in Chess Compositions
114
In Figure 4.9(b), there are 9 pieces and the position would once again not be considered
crowded. It might even be considered equally sparse to 4.9(a), given the slight increase
in pieces. Figure 4.9(c), however, has 23 pieces and is clearly cluttered. Nonetheless,
this position is cluttered not only because of the number of pieces, but also their
proximities to each other. It could be improved by spreading the pieces further apart as
shown in Figure 4.9(d). A position like Figure 4.9(c) can therefore be improved not just
by reducing the number of pieces but also by moving them to locations that would
reduce the overall perceived density. A cluttered or crowded game state would not
entirely be the fault of the number of pieces on the board but also the placement of those
pieces in relation to each other (i.e. the configuration). Figure 4.10 presents a different
perspective.
XABCDEFGHY
8-+-tR-+-+(
7+-+-+-+-'
6-+-+-+-+&
5+-+-+-+-%
4-+-+k+-+$
3+-mK-+-+-#
2-+-+-+-+"
1+-+-+-+-!
xabcdefghy
XABCDEFGHY
8R+-+-+-+(
7+-+-+-+-'
6-+-+-+k+&
5+-+-+-+-%
4-+-+-+-+$
3+-+-+-+-#
2-+-+-+-+"
1+-mK-+-+-!
xabcdefghy (a) (b)
Figure 4.10 Sufficient Sparsity (Constructed Positions)
Neither 4.10(a) nor 4.10(b) would be considered crowded even though 4.10(b) is
arguably sparser given the same number of pieces. However, 4.10(b) is not sparser in a
way that matters to the observer (i.e. that effectively unclutters it). The challenge
therefore lies in creating an evaluation function that is able to clearly discern in terms of
sparsity between certain positions (e.g. Figure 4.9(a) and Figure 4.9(c)) yet classify
115
others as being sufficiently sparse (e.g. Figure 4.10(a) and Figure 4.10(b)). It should
also evaluate some as approximately sparse (e.g. Figure 4.9(a) and Figure 4.9(b)).
4.7.2 A Look at Possible Approaches
One method of evaluating sparsity is to imagine the pieces as pixels on an image grid. A
larger concentration of pixels in a small area of the grid would imply higher density
(Woodruff, 1999). The grid can be divided into quadrants for this purpose and easily
analyzed. However, this method does not translate well to board games because with
only a few pieces, it is not expected that they be evenly spaced out (e.g. one in each
quadrant). That would be unrealistic of most actual game positions. In fact, a position
with a few pieces as little as one square apart (but in a single quadrant), may not be
considered crowded in the context of the game. It might be expected that simply
spreading the pieces further apart increases the overall sparsity of the position but that
would inadvertently consign positions with more pieces to higher densities (there is
nowhere left for the pieces to go).
Dividing the board into smaller sections complicates things further because the ‘average
density’ would then demand that pieces be more evenly spread out across the board in
order to be considered sparse. Both these methods (including other variations of them)
were tested on a few positions and failed to reflect their perceived sparsity. Perhaps the
biggest problem was finding a method that could classify positions like Figure 4.10(a)
and Figure 4.10(b) as being ‘sufficiently sparse’. An arbitrary threshold value for said
classification for example, precluded the use of larger or smaller boards. Furthermore,
using different thresholds for different boards was considered neither practical nor
elegant even though the scope of this research is limited to the 8x8 chessboard.
116
It could be argued that there are perhaps more sophisticated or intuitive ways to gauge
sparsity. For example, the average Chebyshev or Euclidean distance between each piece
and all the others could be used to calculate the ‘average of average distances’ between
pieces to reflect sparsity (a conference paper reviewer suggested this alternative). While
this seems more intuitive and generic it does not in fact, work very well. Using the
Chebyshev distance, Figure 4.9(a) for example, would score 3.43 and 4.9(c), 3.32.
These do not adequately reflect the comparative sparsity of those positions (ignoring
arbitrary or derived multipliers). This approach is more thoroughly tested in subsection
4.7.4(b). A position suffers aesthetically when the pieces are too close together.
Alternative functions – preferably no more computationally intensive than the one
proposed in the following subsection - would also have to ensure that the concept of
‘sufficient sparsity’ is captured. The most widely spread out positions are not
necessarily the most beautiful. Only the cluttered ones are not as beautiful (Troyer,
1983; White, 2003). A system could be trained using supervised learning to recognize
sparsity but that approach would require a reliable (but unavailable) training data set
which would have to be based on human player perception. In addition, it would be
difficult to reconcile the generated function with the concept of sufficient sparsity
explained earlier.
4.7.3 The Sparsity Evaluation Function
The formalization proposed for sparsity is shown in equation 4.7. It is based on the
principle that many pieces surrounding another piece make the area appear dense. The
function was designed primarily with the game of chess in mind.
117
( )( )1
17
1. 1n
nP n s p−
− = + ∑ (4.7)
s(pn) = the number of pieces in the field of a particular piece
The average number of surrounding pieces provides a better idea of how sparse or dense
the overall position is. The reciprocal is used to correlate higher scores with increased
sparsity, and the denominator is incremented by 1 to prevent a division by zero error in
positions of sufficient sparsity (i.e. no immediately surrounding pieces). This also limits
such positions to a maximum score of 1. Arbitrarily moving, adding or subtracting
pieces will not necessarily affect the evaluation score. It also depends on how the
configuration of the position is affected. Given the scope of analysis, evaluation takes
place only in the initial position, and does not encumber the beauty of the combination
itself. The pseudocode for this function is shown in Appendix E.
Figure 4.11 shows the sparsity score for some positions taken from tournament chess
games. Figure 4.11(a) shows a typical middle game position which has 26 pieces that
are quite close together. Its sparsity score is relatively low. Figure 4.11(b) has half as
many pieces (i.e. 13) and is visibly sparser than 4.11(a). Its sparsity score is much
higher given also the different board configuration. The reader should be careful not to
misinterpret the proportional increase in the score which seems to correlate with the
number of pieces, albeit not perfectly.
Figure 4.11(c) shows a modified version of 4.11(b) using the same pieces but with the
board configured to appear less sparse (i.e. more crowding). The score in 4.11(c) is now
comparable to 4.11(a). Figure 4.11(d) shows a modified version of 4.11(a) with the
pieces reduced by about 30% (to 18) and spread out more. The score is about double the
original position. These examples demonstrate how the formalization captures the
118
principle of sparsity and attributes scores based not only on the number of pieces, but
how they are arranged on the board.
XABCDEFGHY
8-+q+-+k+(
7+pzp-zp-vlp'
6-+nzpp+r+&
5zp-+-+-sNQ%
4-+L+P+-+$
3+-zP-+-+P#
2PzP-sN-zPP+"
1+-+R+RmK-!
xabcdefghy
XABCDEFGHY
8-+-+-+Q+(
7zpp+-+R+-'
6-+-+-+n+&
5+P+-zp-+-%
4-+-+-+-mk$
3+-zPq+P+-#
2P+-+-+-+"
1+-+-+-mK-!
xabcdefghy (a) Sperhake vs. Seidler Buchholz sim 4th, 1987
Score: 0.302
(b) Backwinkel vs. Franke Bundesliga 8788, 1988
Score: 0.619 XABCDEFGHY
8-+-+-+-+(
7zpp+-+R+-'
6PzP-+-zpQ+&
5+-+-+-+n%
4-+-+-+-mk$
3+-+-wqP+-#
2-+-+-+KzP"
1+-+-+-+-!
xabcdefghy
XABCDEFGHY
8-+q+-mk-+(
7zP-+-+-+p'
6-zp-zp-zp-+&
5zp-+-+-+Q%
4N+L+PzP-+$
3+-+-+-+P#
2PzP-+-tR-+"
1+-+-+-mK-!
xabcdefghy (c) Backwinkel vs. Franke
Bundesliga 8788, 1988 (Modified Position)
Score: 0.317
(d) Sperhake vs. Seidler Buchholz sim 4th, 1987
(Modified Position) Score: 0.600
Figure 4.11 Sparsity Scores of Chess Positions from Tournament Games
The evaluation function proposed is also theoretically applicable to other board games
since sparsity is a visual property not necessarily related to the rules of the game. Figure
4.12 shows two positions from the game of Go (using a 9x9 board) and their sparsity
scores.
119
(a) Score: 0.214 (b) Score: 0.163
Figure 4.12 Sparsity Scores of Go Positions
It is important not to assess just the empty spaces between the pieces but also the
number of pieces that are on the board and the densities of areas where they cluster. The
evaluation function achieves this by focusing not on the amount of empty space but
rather the concentration of pieces around each other, which leads to the perception of
density. Sufficiently sparse positions therefore are simply those with at least one empty
square between all the pieces (e.g. Figure 4.10(a) and Figure 4.10(b)). While this might
seem like an inherent ‘limitation’, it is more of a slight compromise to capture
realistically the concept of sparsity as it occurs in actual board positions, not
hypothetical ones. The following subsection illustrates this.
4.7.4 Validation
Three experiments were performed for validation of the sparsity evaluation function. It
was tested using a computer program the author developed (see Appendix D).
120
4.7.4(a) Sparsity and Piece Count
The first experiment used 1,000 randomly selected positions from tournament chess
games between expert players (Mega Database, 2008). This was to ensure the games
being tested were not played haphazardly. Assuming that the amount of cluttering
across expert games is generally consistent, it was expected that there would be a
reduction in sparsity scores as the pieces increased. It stands to reason that with fewer
pieces on the board, they are less likely to crowd around each other. The results are
shown in Figure 4.13.
Figure 4.13 Sparsity Values of 1,000 Random Game Positions
The chart shows that sparsity scores based on the proposed evaluation function correlate
negatively with the number of pieces on the board. This (Pearson) correlation was
statistically significant (2T, SL 1%); r = -0.711, t(998) = -31.94, P<0.01. It is important
to note, however, that the correlation is not perfect. The result is consistent with what
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20 25 30 35
Scor
e
Pieces
Positions
121
players would theoretically perceive in general board positions as the game progresses
from the opening to the endgame. As pieces reduce, the board becomes less cluttered.
The chart also shows that positions with the same number of pieces can score quite
differently based on their configurations. This naturally reduces as the board becomes
more crowded, as evident in the chart. The widest range of scores was for 6 pieces i.e.
between 0.333 and 1 and the lowest for 30 pieces, between 0.254 and 0.263. It is
notable that only 5 positions from the 1,000 tested achieved the sparsity threshold (i.e.
the maximum score of 1) and these had between 4 and 6 pieces.
4.7.4(b) Sparsity and Piece Count (Alternative Method)
Given that there are probably many ways to evaluate sparsity, the same experiment was
performed on the same data using the alternative method of ‘average of averages’ (see
subsection 4.7.2). The results are shown in Figure 4.14. The scores correlated poorly
with the piece count and positively at that (2T, SL 1%); r = 0.465, t(998) = 16.59,
P<0.01. Even so, the differences in configuration appear to be captured here as well.
If aesthetic perception of sparsity is the goal, a threshold of ‘sufficient sparsity’ is
necessary because many positions would not be considered cluttered enough to warrant
improvement in that regard (see Figure 4.10). This can be incorporated into the design
of the function itself (as per the one proposed, see equation 4.7), arbitrarily specified (as
a multiplier) or perhaps through the use of fuzzy logic.
122
Figure 4.14 Sparsity Values of 1,000 Random Game Positions (Alternate)
The aim of this experiment was to illustrate that seemingly obvious ‘better’ alternatives
to the function proposed might not actually work as well in practice. It is fallacious to
assume that they are necessarily better simply because they appear more complex or
intuitive than the one proposed, which was deemed sufficient for at least the purposes of
this research.
4.7.4(c) Sparsity and Piece Configuration
A third and final experiment compared two sets of chess compositions (Meson Chess
Problem Database, 2008) against two sets of real games to see if a difference in
configuration was really being gauged by the function proposed. All four sets consisted
of approximately 100 randomly selected positions each (regrettably the original sets,
including knowledge of their exact sizes, were lost by the time of writing; 100 positions
per set is an educated guess based on the author’s usual style of analysis and degrees of
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
0 5 10 15 20 25 30 35
Scor
e
Pieces
Positions
123
freedom in the original statistical results reported below). Compositions typically
feature unnatural positions with distinct configurations.
No assumption was made about which domain (compositions or real games) should
have greater sparsity even though cluttering is best avoided according to composition
convention (see subsection 3.3.1). However, a difference in average sparsity was
expected between these domains. The initial positions of orthodox direct mate-in-3
compositions were compared against initial positions of mate-in-3 combinations taken
from expert tournament games. All positions contained only 12 pieces, which is
incidentally the upper limit of ‘Meredith problems’ (that contain anywhere between 8-
12 pieces). This was considered a suitable common ground for the experiment.
There was no significant difference (TTUV, 2T, SL 5%) between the sparsity score
means of the real game sets (i.e. 0.45986 and 0.46035) but there was a small yet
significant difference of 0.0544 between the means of the composition sets (i.e. 0.4279
and 0.4823); t(190) = -5.2, P<0.01. This is not entirely unexpected – even between
randomly selected sets – because compositions are created by humans (more creativity
is usually involved here than in real games) and tend to have original configurations.
Real games are therefore less likely to show a significant difference in this sense.
Even so, any significant difference in means between two randomly selected sets from
the same domain can still be attributed to random factors (although this might call into
question the validity of the evaluation function used). Both pairs of composition vs. real
game sets (i.e. COMP Set 1 vs. RG Set 1, and COMP Set 2 vs. RG Set 2) showed small
yet significant differences between their means (TTUV, 2T, SL 5%); t(190) = -3.0,
P<0.01 and t(197) = -2.3, P<0.05, respectively. The results therefore indicate that piece
124
configuration was also being gauged by the evaluation function proposed since piece
count was the same in all sets.
An experiment seeking correlation between sparsity scores and the perception of human
subjects was avoided because it would be difficult for them to tell apart small
differences in sparsity between positions. There were also resource limitations and
concerns about leading questions as described earlier (see subsection 4.5.5(c)).
4.7.5 Discussion
The experimental results in the previous section suggest that the proposed evaluation
function is able to assess sparsity in the game of chess and perhaps similar board games.
Its mathematical simplicity also minimizes computational load. Sparsity alone may not
make a position beautiful but it is a universal aesthetic factor regardless of board size or
piece type. Game rules might influence perception of sparsity (e.g. in the board game
‘Amazons’ all the pieces are long-range) but this remains to be investigated and is
beyond the scope of this thesis. There could also be differences due to playing strength.
For example, stronger chess players tend to perceive more squares with a single eye
fixation and spend less time inspecting each square than weaker players (Blignaut et al.,
2008).
The main concern here, however, is purely visual in nature and relates primarily to the
game of chess. It could even be that the inherent mobility of the different piece types
plays a role in the perception of sparsity. This means for example, that a knight
surrounded by pawns is considered sparser than a queen surrounded by pawns (also
suggested by a conference paper reviewer). However, this assumption has no basis in
125
the literature surveyed which suggests instead that proximity of the pieces is the
essential factor in connection with cluttered positions in chess (Troyer, 1983; White,
2003). Based on the experiments performed in the last section, the evaluation function
proposed (see subsection 4.7.3, equation 4.7) was therefore deemed suitable for at least
the purposes of this research.
4.8 Points of Evaluation for the Aesthetic Principles
The following table lists a summary of the ‘points of evaluation’ (see section 3.8) in a
combination for each aesthetic principle explained in this chapter, given the scope of
analysis (see section 3.7).
Table 4.1 Points of Evaluation for the Aesthetic Principles
Aesthetic Principle Point(s) of Evaluation 1 Violate Heuristics Successfully 1 2 Use the Weakest Piece Possible 3 3 Use all of the Piece’s Power 1, 2, 3, 4 4 Win with less Material 0 5 Checkmate Economically 4 6 Sacrifice Material 4 7 Spread out the Pieces (Sparsity) 0
These are the points or stages in a mate-in-3 combination where each aesthetic principle
is computationally evaluated using its proposed formalization. For a more detailed
explanation, refer to the individual sections above.
4.9 Chapter Summary
This chapter explained in detail the logic behind the development of each formalization
(i.e. evaluation function), for each of the seven selected aesthetic principles. The first
126
aesthetic principle of ‘successfully violate heuristics’ consisted of four heuristics
namely ‘keep your king safe’, ‘capture enemy material’, ‘do not leave your own pieces
en prise’ and ‘increase mobility of your pieces’. Along with the second, third and fourth
principles (see Table 4.1), their formalizations were comparatively straightforward. The
fifth principle, i.e. ‘checkmate economically’ was more complex and therefore divided
into eight sections containing, notably, a computer algorithm and a section on
validation. It is notable that there is a difference between the second (i.e. use the
weakest piece possible) and fifth aesthetic principle despite the former relating to the
checkmating piece (given the scope of analysis, see section 4.2). This is because the
second principle has nothing to do with the board configuration or other pieces, quite
unlike the fifth principle.
The sixth aesthetic principle, i.e. ‘sacrifice material’ was presented next. It was
explained that the principle could account for the ‘clean’ value of all material sacrifices
in a combination, even taking into account pawn promotions. Finally, the seventh
principle (i.e. ‘sparsity’) was explained. Similar to the principle of checkmating
economically, this was also relatively complex. It was divided into five sections,
including one on validation. The proposed formalizations (i.e. evaluation functions) for
all these seven aesthetic principles form part of the cumulative aesthetic assessment in a
combination (see section 3.5). The next chapter details the formalizations that were
developed for the ten selected themes.
127
CHAPTER 5: METHODOLOGY – Theme Formalizations
5.0 Formalizing the Ten Themes
Chess themes are an integral part of the game beyond its basic rules. Players not only
use themes to improve their games but also appreciate them aesthetically (Averbakh,
1992; Seirawan, 2005). Chess compositions for example, employ a range of themes
primarily for aesthetic reasons. Even in real games, master players would often rather
win a beautiful and thematic game than simply by any means possible (Kasparov, 1987;
Joireman et al., 2002; Damsky, 2002).
This chapter details the formalizations developed by the author for the ten selected
themes (see section 3.4) to estimate their aesthetic value. The themes are as follows.
1. fork
2. pin
3. skewer
4. x-ray
5. discovered/double attack
6. zugzwang
7. smothered mate
8. cross-check
9. promotion
10. switchback
The general methodology of development (see section 3.6) is more easily applicable
here than with the aesthetic principles (see chapter 4) because the ideal example of a
theme in most cases can be represented using a single benchmark value. Some themes,
however, like the zugzwang, ‘smothered mate’, cross-check and ‘pawn promotion’ are
less complex than others. ‘Liabilities of consequence’ (i.e. manoeuvres by the opponent
that have a prominent effect on the strengths of a theme) for them, could not be found or
128
were deemed irrelevant. The variety of themes and the logic behind their formalizations
should present the reader with a good overview about how evaluation functions for
other themes can also be developed. Unless otherwise mentioned, the chess positions in
the diagrams are constructed ones. This was done to illustrate the themes properly.
Positions from real games and compositions are also used to illustrate actual examples
of how the aesthetic scores for certain themes are calculated (Mega Database, 2008;
Meson Chess Problem Database, 2008). For practical purposes, pawns are not included
as thematic pieces (i.e. those considered as being part of a theme) for the fork, pin,
skewer and ‘discovered/double attack’ (see subsection 4.1.2). They are nonetheless a
valid liability in the form of a piece that could intervene between two others (see section
3.6).
A single piece (e.g. queen) capable of intervening on two different squares on a line
counts as two interventions by default. Blocks of pseudocode, showing programmatic
implementation of some of these functions, are provided in Appendix E, for reference.
For the equations, v() denotes the value of a piece, r() its power, d() the distance
traversed by a piece in a move or between two piece and i() the number possible
intervening pieces between two others. v(i()) refers to the value of those pieces. This
information is presented here to avoid repetition.
5.1 Fork
The fork (Figure 5.1(a)) can be defined as a direct and simultaneous attack on two or
more enemy pieces by any single piece (Hooper and Whyld, 1996). The theme is often
performed using the knight due to its unique and wide-range L-shaped movement (see
129
Appendix A, Figure A.5). A two-pronged fork is the most common. A knight or queen
can theoretically fork up to 8 pieces. In terms of effectiveness, the main features that
differentiate one fork from another include the types of pieces forked (i.e. their values),
the number of prongs used (equivalent to the number of forked pieces) and the lack of
defensive manoeuvres by the opponent (i.e. ‘liabilities’, see section 3.6) such as possible
checking moves. In terms of beauty, the distances between the forking and forked
pieces (it reflects more power of the former being used) and economy (e.g. using a
weaker piece to fork) play a role as well (Margulies, 1977; Levitt and Friedgood, 2008).
XABCDEFGHY
8-+-+-+-+(
7+-+-+q+-'
6-+k+-+-+&
5+-+-sN-+-%
4-+-+-+r+$
3+-mK-+-+-#
2-+-+-+-+"
1+-+-+-+-!
xabcdefghy
XABCDEFGHY
8R+-+-+-+(
7+-+-+-+k'
6-+-+LzP-+&
5+-+-+-zPp%
4-tr-+-+-zp$
3+-+-+-+r#
2-+-+-+-zP"
1+-+-+-mK-!
xabcdefghy
(a) (b)
Figure 5.1 The Fork
Forking a king and queen (sometimes described as a ‘royal’ fork) is arguably better than
forking a rook and bishop because in the former case, cumulative material value is
higher (i.e. the threat is greater). Doing so with a bishop would also be better
thematically than with a queen because the bishop is more economical and there is less
material being put at stake on the part of the one executing the theme. The number of
prongs indicates how many different pieces are forked. It might therefore seem that a
four-way fork is better than a two-way fork but this is not necessarily so. In terms of the
threat created and material to be won, forking say, two rooks would still be better than
130
four pawns. However, more prongs being used can be construed as less wastage of the
forking piece’s power (economy). The aesthetic score for the fork theme, T1 can be
calculated using equation 5.1 below.
( ) ( ) ( )( )( )111
1 1. , .n n
c n k n kT f v fp n d f fp r f k−− = + + − ∑ ∑ (5.1)
fc = fork constant (i.e. 37), fp = forked piece, fk = forking piece,
k = number of checking moves by fp
The benchmark for the fork, fc was determined by first selecting the average number of
possibly forked pieces (i.e. between 2 and 8) which is 5. The value of the most valuable
pieces on the chessboard that could be forked in that way (assuming only the original
set of pieces) namely the king, queen, two rooks and a bishop was then summed and
added to the corresponding number of prongs required (5) for a total of 37. The absolute
maximum of 8 forked pieces was not used because this is extremely unlikely.
When a fork occurs, the value of the forked pieces and prongs used, n is first added
together. The sum of ‘distance to piece power’ ratios between the forking piece, fk and
each of the forked ones, fpn is then added to that value. This ratio differentiates between
pieces of different strength that move the same distance (see section 4.3). Using the
average distance was considered unsuitable because the average value of forked pieces
was not used. Averaging in this way tends to represent the actual pieces involved in the
theme and their applied powers inadequately.
Possible checking moves, k by the forked pieces is considered a liability and subtracted
from the total. If the opponent’s king happens to be one of the forked pieces, the
likelihood of k is much lower because the threat must be dealt with immediately unless
131
there is the possibility of a cross-check (see section 5.8). A higher number of checking
moves increases the liability to the theme. Possible intervening pieces were not taken
into account because the fork typically has multiple threats and the effect of such pieces
is not as prominent as it is in themes where there is only one threat. As mentioned in
subsection 3.8.1, theme detection should apply only to the piece that moved last. This is
to avoid evaluation of ‘discovered’ forks that do not have the same effect since at least
one enemy piece would have had to have been already under attack by a piece that did
not move.
There are some positions in which a move could also ‘activate’ a fork, even without
either forked piece threatened just prior. Re4 in the position (FEN:
2b4r/p7/6p1/1P1pkpP1/3R1R1P/3Br3/2P3K1/8 b - -) taken from the game (N.N. vs.
Mannheimer, Frankfurt am Main, 1921) does not register as a fork even though it cuts
off the white rooks from each other and activates the fork by the black king on e5 (see
Appendix C, Figure C.8). A variation of a fork already in effect may also be repeated
(e.g. by moving a single square) but technically this still qualifies because the piece is
no longer on the same square. In the constructed position (FEN:
4k3/8/1nn5/1Q6/8/8/8/K7 w - -), the move, Qc5, still qualifies as forking both knights
(see Appendix C, Figure C.9).
One of the peculiarities in chess that was discovered by the computer program designed
for this research (see Appendix D) can be seen in Figure 5.1(b) where the bishop has
just moved from d5 to e6. Since ‘mating squares’ are also considered legitimate items
that can be forked, this move qualified by threatening occupation of the f5 square and
also the rook on h3. It is not a typical fork since only one line is involved and the rook
is attacked through the mating square but the threat is similar. There was nothing in the
132
literature surveyed to exclude this type of position from being perceived as a fork so it
was not invalidated.
Such a fork, however, is considered to have two prongs. Even so, multiple (empty)
mating square threats along the same line should be limited to just one square.
Otherwise, this would score unnecessarily high aesthetically for positions where
checkmate could be delivered on say, any three adjacent squares on a line. If the king
happens to be one of the forked pieces, mating squares do not count because the king
itself is already threatened with capture. For reference, the forking move Ne5+ in Figure
5.1(a) scores 0.75 aesthetically as shown in the sample calculation below.
( )( )1 11 37 24 3 6 8 0 0.75T − − = × + + × − =
Be6 in Figure 5.1(b) scores 0.441. The pseudocode for this function is shown in
Appendix E.
5.2 Pin
A pin is in effect when a long range piece (i.e. bishop, rook or queen) attacks an enemy
piece and prevents it from moving lest the more valuable or undefended piece behind it
be captured. Figure 5.2 shows the three pieces involved in a pin. The ‘pinning’ piece is
the bishop, the ‘pinned’ piece the knight, and the ‘target’ piece the king. This is
generally known as an ‘absolute pin’ because it is illegal for the knight to move (Hooper
and Whyld, 1996). Absolute pins are often the strongest examples of this theme.
133
XABCDEFGHY
8-+-+-+-+(
7+-+-+-+k'
6-+-+-+-+&
5+-+-+-+-%
4-+-+-+-+$
3+-+n+-+-#
2-+L+-+-+"
1+-+K+-+-!
xabcdefghy
Figure 5.2 The Pin
A ‘relative pin’ occurs when the target piece is simply more valuable than the pinned
one (king excluded) whereas in a ‘partial pin’, the pinned piece is long-range and can
still move along the pinning line. Finally, a ‘situational pin’ is one where the target is a
mating square or a square where occupation would be highly disadvantageous for the
opponent. The main features that differentiate one pin from another include the values
of the pieces, distances between them and the mobility of the pinned piece. Liabilities
include possible checking moves and intervening pieces. The formalization used to
evaluate the aesthetic score for a pin, T2 is shown in equation 5.2.
( ) ( ) ( ) ( )( ) ( ) ( )( )( ) ( ) ( )( )
1 112
1
. , . . ,
, . , , 1
0, 0
c p t n t n p p a
n p n n pa
T p v p v p d p p r p m p r p k l
i p p v p v i p p ili
− −−+ +
−
= − + + + ≥ = =
∑ (5.2)
pp = pinned piece, pt = target piece, pn = pinning piece,
k = number of possible checking moves by pp and pt, la = (additional) liabilities,
pc = pin constant (i.e. 19)
The values of the pinned and target pieces are first summed. The distance between the
pinning and target piece (taken as a ratio against the piece power of the pinning piece) is
134
then used because this is the full length of the theme. The second part of the
formalization includes liabilities to the theme. For example, in a partial pin, the ratio of
the pinned piece’s movable squares, m(pp) against its piece power is used since
complete immobilization would be ideal. ‘Movable squares’ pertaining to the pinned
piece are those it can move to without exposing the target piece, but excluding the
square of the pinning piece (i.e. capturing it), because the pin manoeuvre is already
deemed favourable for White.
The number of possible checking moves, k by the pinned and target pieces are also a
liability. Possible intervening pieces between the pinned and pinning pieces, if any, are
evaluated in terms of their number and value in relation to the latter. More intervening
pieces and those of lesser value increase the liability. This is because the possibility of
intervening with a pawn (making it prone to capture) to deal with the pin is less risky
for the opponent than using a queen. The possibility of using several pawns typically
reduces the risk even further. The advantage of aesthetic analysis is that one does not
have to consider other positional implications of these possible responses because the
result of the move combination is already known, making them irrelevant.
If the achievement of the combination (e.g. checkmate) or theme (e.g. winning material)
is unknown or cannot be determined, aesthetic evaluation is of little value. This is why
the context in which these formalizations are applied is important. If a theme is
evaluated on its own outside the context of a combination, then a new goal such as
material gain or positional superiority should be determined first. This does not mean
ensuring that the best possible moves were played but simply that the result of the
moves played is tangible.
135
The pin constant (i.e. 19) was derived from what would arguably be the ideal pin (queen
against king) with no liabilities. Even though this is not quite possible on the chessboard
given the queen’s nature as a long range piece herself (she will most likely have some
mobility along the pining line despite the pin), what is possible is approximately
equivalent, especially if a discovered attack on the pinned piece impedes its mobility
further. The constructed position (FEN: 3bkn2/KP6/1N2q1P1/8/7B/8/4B3/4R3 w - -) is
a good example. Bc4! pins the queen against the mating square on f7 and
simultaneously uncovers another pin on the queen by the rook on e1 which limits her
mobility. This scores 1.012 (see Appendix C, Figure C.10). A mating square in the
context of a pin only counts if the pinned piece cannot move to nullify that status.
In chess it is possible for two-way or more pins to occur (using the same pn). In such
cases, the value is calculated for each pin and then summed. Bxd5! in the constructed
positions (FEN: k5r1/1q3n2/8/3p4/8/1n3B2/r7/3R1K2 w - -) scores 1.719 for the three-
way pin. Computational detection should therefore not stop at the first pin found (see
Appendix C, Figure C.11).
Pins that were already in effect (using the same pieces) prior to the move do not qualify
for assessment. Unlike the fork (see previous section), these themes do not inherently
have multiple threats so a repeated configuration can be excluded without the ambiguity
of determining to what degree the repetition occurred. The following position taken
from a tournament game demonstrates how the aesthetic score of the pin is calculated.
136
XABCDEFGHY
8-+-+-+-+(
7zp-+-+-trk'
6-sn-zp-+rzp&
5+-+-+-+-%
4-+-zPQ+-+$
3+-+-zP-+-#
2PzPP+-+-+"
1+-mK-+-+R!
xabcdefghy
Porth vs. Wihl, Baden-Baden op (1), 1987
Figure 5.3 Aesthetic Assessment of the Pin
In Figure 5.3, the queen has just moved to e4 and pins the rook on g6 against its king.
The rook has no mobility along the pinning line (or any other line for that matter since
this is an ‘absolute pin’). There are also no checking moves possible or intervening
pieces. Programmatically, it is important to determine if a piece is pinned against its
king in the position because simple move capability testing might (e.g. in this position)
result in a false positive for possible checking moves (e.g. Rg1+). The aesthetic score
calculation for the pinning move played in Figure 5.3 is as follows.
( ) ( )1 1 12 19 5 10 3.27 0 14 0 0 0.795T − − − = × + + − × + + =
A negative evaluation score with the pin formalization is possible but uncommon. It
occurred in 1 out of 1,139 pins detected in the author's collection of 31,896 composition
and tournament game mating combinations. This was determined with the aid of the
computer program developed for this research (see Appendix D). In the position (FEN:
Q1K5/4B3/1p6/4kpp1/3N1N2/P2p4/rb6/8 w - -) taken from the #3 composition
(Williams, P.H., Reading Observer, year unknown); after 1. Qg2, the queen - while
137
technically pinning the bishop on b2 against the undefended rook - makes itself prone to
a discovered attack should the bishop move. Additionally, a pawn can intervene on d2
(score: -0.094). Evaluation of the pin is based primarily on the thematic pieces and not
the position at large or composition as a whole. In principle, this is a poor example of
the pin (see Appendix C, Figure C.12). The pseudocode for this function is shown in
Appendix E.
5.3 Skewer
The skewer is like an inverse pin (Figure 5.4). The more valuable piece is the one
immediately attacked or ‘in front’. In the diagram, the king is skewered against his
queen. If both enemy pieces have the same value, it is still a skewer, not a pin. To
ensure skewers and pins are mutually exclusive, pins are restricted to where the target
piece (i.e. the one ‘behind’) is worth more than the pinned one.
XABCDEFGHY
8-+-+-+-+(
7+-+q+-+-'
6-+-+-+-+&
5+-+-+-mK-%
4-+-+-+-+$
3+-+k+-+-#
2-+-+-+-+"
1+-+R+-+-!
xabcdefghy
Figure 5.4 The Skewer
The formalization used to evaluate the ‘aesthetics of the skewer’, T3 (shown in equation
5.3) is similar to the one used for the pin (see equation 5.2). The details of this
138
formalization are the same as what was explained in section 5.2 (for the pin) and
therefore not repeated here.
( ) ( ) ( ) ( )( ) ( ) ( )( )( ) ( ) ( )( )
1 113
1
. , . . ,
, . , , 1
0, 0
c p t n t n p p a
n p n n pa
T s v s v s d s s r s m s r s k l
i s s v s v i s s ili
− −−+ +
−
= − + + + ≥ = =
∑ (5.3)
sp = skewered piece, st = target piece, sn = skewering piece,
k = number of possible checking moves by sp and st, la = (additional) liabilities,
sc = skewer constant (i.e. 19)
The ideal skewer that serves as a benchmark for this theme would be a bishop
skewering the king against his queen at the furthest distance with no liabilities. A
skewered piece that may be able to come to the immediate defence of the target piece
after moving off the skewering line does not affect the theme’s validity because of the
analysis ply depth (see section 3.8). The skewer is typically a more powerful tactic than
the pin because the skewered piece is compelled to move or be captured.
XABCDEFGHY
8-+Q+kvl-+(
7+-+-+p+-'
6-wqrzp-zP-tr&
5zp-+-+-+p%
4Lzp-+P+-+$
3+P+-+-+-#
2-zPP+-+-zP"
1+K+R+-+-!
xabcdefghy
Gutierrez vs. Hernandez, Caracas, 1973
Figure 5.5 Aesthetic Assessment of the Skewer
139
In Figure 5.5, the queen has just moved to c8, skewering the king against the bishop on
f8. The rook on c6 is unable to capture the queen because it is pinned. The enemy queen
is the sole possible intervening piece. The aesthetic score of the skewer is calculated as
shown below.
( ) ( )( )1 1 1 13 19 10 3 3 27 0 10 0 1 9 9 0.585T − − − − = × + + × − × + + + × =
5.4 X-Ray
The x-ray theme (Figure 5.6(a)) is similar to the pin and skewer except that the
opponent’s piece is between two friendly long-range pieces (capable of defending each
other) and can capture either one (Silman, 1998).
XABCDEFGHY
8-+k+-+-+(
7+-+-tR-+-'
6-+-+-+-+&
5+-+-+-+-%
4-+-+q+-+$
3+-+-+-+-#
2-mK-+R+-+"
1+-+-+-+-!
xabcdefghy
XABCDEFGHY
8-+-+k+-+(
7+-wQ-+-+-'
6-+-+-+-+&
5+R+-wq-+R%
4-+-+-+-+$
3+-+-+-vL-#
2-+-+-+-+"
1+-+K+-+-!
xabcdefghy (a) (b)
Figure 5.6 The X-ray
It is more of a defensive tactic since the x-rayed piece (in the middle) would have had to
have been under attack by at least one of the x-raying pieces in the move prior. At first
glance, it would seem that a more highly valued x-rayed piece is better but aesthetically,
140
the opposite is true due to the theme’s paradoxical nature. Unlike the previous themes,
in an x-ray the opponent’s piece is already in a position to be captured but in fact, is not
for whatever reason. Instead, a defensive manoeuvre is performed which ‘defends’ the
supposed attacker through the enemy piece by placing another one in jeopardy. This
removes the emphasis from material gain and economy to paradox. The formalization
developed for this theme, T4 is as shown in equation 5.4.
( ) ( ) ( )( ) ( ) ( ) ( ){ }( )114 1 2 1 2 1 2. 1 , . min ,c rT x v x v xp v xp d xp xp r xp r xp
−− = + − + +
(5.4)
xr = x-rayed piece, xp1 = x-raying piece 1,
xp2 = x-raying piece 2, xc = x-ray constant (i.e. 7)
The distance between x-raying pieces is divided by their average power since two are
necessarily involved – the order is not important - and they could be of different types.
Both ends of the configuration spectrum (i.e. two queens x-raying a bishop and vice
versa) are of approximate value due to their paradoxical nature. The x-ray constant (to
the nearest integer) is such that these configurations score approximately 1.
Combinations of other pieces score between 0 and 1.
Intervening pieces and checking moves are irrelevant because the x-rayed piece was left
there to begin with. An opponent piece can be x-rayed multiple times (Figure 5.6(b)) but
this is accounted for as two separate instances of the theme in two different moves. A
single piece that x-rays two opponent pieces simultaneously is also possible. In the
constructed position (FEN: K6B/B5q1/8/2q5/p2n4/k1P5/8/6B1 w - - 0 1), the move
Bxd4 x-rays both the enemy queens on c5 and g7 (score: 2.077). A manoeuvre like this
is highly uncommon (see Appendix C, Figure C.13).
141
A notable programming issue in detecting legitimate x-rays on the chessboard is testing
for the defensive capabilities of the x-raying pieces. While long range pieces (i.e. queen,
rook and bishop) are the only ones to be tested for, in pairs their control lines are
limited. Table 5.1 shows these limitations so validity of the theme can be determined
programmatically at the start. By following the general development methodology (see
section 3.6), time and computer processing power can be saved by not having to test for
pieces that are irrelevant to the theme.
Table 5.1 X-ray Defensive Capabilities
xp1/xp2 Q R L Q RFD RF D R RF RF X L D X D
R = rank, F = file, D = diagonal, X = none
Standard chess programming techniques for determining move legality need to be
adapted when testing for these defensive capabilities (Hyatt, 1999; Tannous, 2007). For
example, the x-rayed piece in the middle can be temporarily removed from the board in
order for said capabilities to register in the program. A piece cannot ordinarily be said to
defend another if the line of defence is not clear (a long range piece cannot ‘jump’ over
another). It is also important to test if the x-rayed piece can potentially capture either x-
raying one. Two rooks x-raying an enemy bishop for example, might register as a valid
instance of this theme based on the types of pieces involved and the defensive capability
of the rooks. In effect, it is not an x-ray because the bishop cannot move along ranks
and files.
142
XABCDEFGHY
8k+n+-+-+(
7+L+-+-+-'
6-+-+N+-+&
5+-+q+-+-%
4-tR-+-+-+$
3tR-+-+-+-#
2-+K+-+-+"
1+-vl-+-+Q!
xabcdefghy
C. S. Kipping, Chess, 1941
Figure 5.7 Aesthetic Assessment of the X-Ray
The position in Figure 5.7 was taken from a chess composition. White has just played
his bishop to b7 and x-rayed the queen on d5 between itself and the queen on h1. The
aesthetic score of the x-ray is calculated as shown below. The pseudocode for this
function is shown in Appendix E.
( ) [ ]( )114 7 1 9 6 6 13 0.637T −−= × + − + × =
5.5 Discovered/Double Attack
The discovered attack is a powerful tactic in chess where moving a piece uncovers an
attack on an enemy piece. In Figure 5.8(a), the knight is the moving piece and the queen
the discovered piece. The black king is the one attacked. The discovered attack becomes
a double attack if the moving piece uses the opportunity to attack another piece or the
same one that is facing the discovered attack.
143
XABCDEFGHY
8-+-+-+-+(
7+-+-+-mk-'
6-+N+-+-+&
5+-+-+-+-%
4-+-+-+-+$
3+-wQ-+-+-#
2-+-mK-+-+"
1+-+-+-+-!
xabcdefghy
XABCDEFGHY
8-+-+-+-+(
7+-+-+-+-'
6-+-mk-+-+&
5+-zpP+-+-%
4-vL-+-+-+$
3+-+-+-mK-#
2-+-+-+-+"
1+-+R+-+-!
xabcdefghy
(a) (b)
Figure 5.8 The Discovered/Double Attack
In the diagram (Figure 5.8(a)), if there was a rook on b8, the knight would not only have
created a discovered check on the king but also a double attack by threatening the rook.
Where the king alone is attacked, it is a discovered check (as per Figure 5.8(a)) or
double check (e.g. Nf5 instead). The latter forces the king to move because both pieces
cannot be captured or intervened against at the same time.
If a double attack involves three or more pieces (e.g. a knight moves to create a
discovered attack and simultaneously delivers a fork on two other pieces), only the more
powerful of the two counts along with the discovered one. The fork should be evaluated
separately. Mating squares were deemed unsuitable (as possible threats) for this theme.
Any piece is capable of uncovering an attack on an enemy one so long as there is a long
range piece behind it. The theme is evaluated as follows (equation 5.5).
144
( ) ( ) ( ) ( ) ( ) ( )( ) ( )
( ) ( ) ( )( ) ( )
1 115
1
. , . , . ,
, . , , 1, 00, 0
c m k m m m k k k a
k k k k ka m
T b v ba v ba d b ba r b d b ba r b k l
i b ba v b v i b ba il v bai
− −−
−
= + + + − + + ≥ = = =
∑ (5.5)
bam = piece attacked by the moving one, bak = piece attacked by the discovered one,
bm = moving piece, bk = discovered piece, k = number of possible checking moves by
bam and bak, la = (additional) liabilities, bc = ‘dda’ constant (i.e. 20)
The theme constant, bc, is derived from the ideal instance of this theme, namely the
double check (twice the value of the king). Possible intervening pieces, if any, are only
a liability if it is a discovered attack (not a double attack) because they are usually not
effective against themes with inherent multiple threats (e.g. the fork, see section 5.1).
The following position was taken from a tournament game as an example of how the
score for this theme is actually calculated.
XABCDEFGHY
8-+-+R+-mk(
7+-+-+-+-'
6-+-+-+-zP&
5+-+-+-+K%
4-+-+-+-+$
3+-+-+-+-#
2L+-+-tr-+"
1+-+-+-+-!
xabcdefghy
Markland vs. Lee, BCF-ch, 1971
Figure 5.9 Aesthetic Assessment of the Discovered Attack
In Figure 5.9, the bishop has just moved from g8 to a2, creating a discovered attack on
the enemy king by the rook on e8. The rook on f2 is the single possible intervening
145
piece. The aesthetic score for the discovered attack (i.e. discovered check) in this move
is calculated as follows.
( ) ( )( )1 1 1 15 20 . 0 10 0 13 3 14 0 1 5 5 0.411T − − − − = + + × + × − + + × =
The pseudocode for this function is shown in Appendix E. There is a very rare instance
of this theme known as a ‘two-way discovered attack’ (Figure 5.8(b)) that is only
possible through an ‘en passant’ capture of the opponent’s intervening pawn (Newborn
et al., 2005; Levitt, 2006). It is equivalent to a double attack. In the diagram, the bishop
(originally at d2) had just moved to b4, placing the enemy king in check. The black
pawn intervened by moving from c5 to c7 but now permits an ‘en passant’ capture on c6
by the white pawn which results in the two-way discovered check by the bishop on b4
and rook on d1. The bishop, as the unexpected secondary discovered piece, assumes the
role of bm (the moving piece) for calculation purposes.
Given its rarity in games and even compositions, programmatic detection of the two-
way discovered attack – which involves additional coding and processing power - is not
essential. For this research, the two-way discovered check (when the opponent piece in
question is the king) was programmed for detection because its effect is the most
prominent. To minimize computational overhead, such an attack on other pieces was not
included. Instead, they are detected and evaluated as regular discovered attacks. A
castling manoeuvre could also incorporate this theme.
Such an instance was unexpectedly discovered by the computer program (see Appendix
D) developed for this research. In the position (FEN: 8/rb2p2P/3p4/5P2/2N5/8/1p6/
k3K2R w K -) taken from the #3 composition (Bottger, H., Die Schwalbe, 2007); after
146
1. h8=Q e5 2. fxe6 e.p. Ra2, White castles and mates with 3. 0-0# (score: 0.518); see
Appendix C, Figure C.14.
By chance, the author happened to construct a piece configuration of this theme that
was apparently not explicitly accounted for during the coding process. This in turn led
to the discovery of other, similar configurations that had not been accounted for. Even
though the formalization proposed (see equation 5.5) is able to account for them, the
computer program was unexpectedly handling configurations like these in its own way
based on the basic structures that had been programmed into it. An example is the
constructed position (FEN: 3k4/8/6r1/8/8/3N4/8/1BKR4 w - - 0 1). After Nf4+, we are
left with the curious double-discovered attack on the enemy king and rook (see
Appendix C, Figure C.15).
It was apparently making decisions about which pieces to evaluate (and which not to)
when encountering more than the expected number of choices. This is most likely a
form of self-organization, which can occur in software (Mamei et al., 2006). Since these
piece configurations and their computations were getting ever more complex, the author
decided to retain the methods arrived at by the computer program. Once they were
deciphered, these methods seemed reasonable, after all.
5.6 Zugzwang
This theme refers to positions where any move puts the player whose turn it is at a
greater disadvantage (e.g. checkmate, loss of material) than if the turn could be skipped.
It usually occurs in the endgame where there are fewer pieces on the board and
therefore also fewer legal moves available. The main factor that differentiates one
147
zugzwang from another is how many moves (all disadvantageous) are available to said
player. A position where it would be disadvantageous for either player to move is called
a ‘mutual’ or ‘reciprocal zugzwang’ (Rice, 1997). The aesthetic score of a zugzwang is
calculated as follows (see equation 5.6).
16 .c mT z z−= (5.6)
zm = the number of legal moves in a position, zc = zugzwang constant (i.e. 30)
The benchmark is based on the average number of moves in a typical chess position
(Shannon, 1950; Allis, 1994). Even if it were known, the maximum number of moves
possible in a legal chess position is not suitable because zugzwangs are rather limited to
positions where moves are below average (Tabibi and Netanyahu, 2002). The aesthetic
value of a zugzwang therefore tends to correlate positively with its complexity and
improbability of occurring (Levitt, 2005). There were no identified liabilities of
consequence to this theme so the proposed formalization sufficed.
In Figure 5.10, the rook has just moved to h1, making all possible moves by the
opponent detrimental to himself (in this case, permitting checkmate). However, should
he be allowed to skip his move, White cannot immediately checkmate. This is what
separates many positions that look like a zugzwang (common in forced combinations)
but are actually not. Programmatically, it can be determined by permitting a null move
(i.e. no move) to Black and looking for a forced mate-in-1. If there is none, the theme is
valid. The pseudocode for this function is shown in Appendix E.
148
XABCDEFGHY
8-+-+-+-+(
7+-+-+p+-'
6-+-+pwQ-+&
5+-+-tRl+-%
4-+-+-+k+$
3+-+P+-+-#
2-+-+-+PmK"
1+-+-+-+R!
xabcdefghy
Figure 5.10 The Zugzwang
In positions where the achievement is not as clear as immediate checkmate but
something like significant material gain or forced mate-in-3, this determination is more
difficult because each variation spawning from each possible response by the opponent
needs to be carefully analyzed. It is likely that even experienced players might perceive
certain positions to be in zugzwang when in truth they are not due to some subtle
defence uncovered through exhaustive computer analysis.
In standard chess playing programs (and other similar games), zugzwang detection can
be a problem because there is no well-defined test criterion for such positions except the
upper or lower bound value produced by its generic evaluation function. These values
typically favour a change of some kind on the board over the status quo (Heinz, 1999b;
Tabibi and Netanyahu, 2002). It may also suffer from the horizon effect where a tactical
threat is missed due to an insufficient search depth (Guid and Bratko, 2007; Nagashima,
2007). Quiescence search (i.e. looking a little further along ‘important’ lines) is no
guarantee against this problem either. In theme (aesthetic) evaluation, a zugzwang is
detected for its own sake and usually with a very specific achievement or goal in mind
that can be determined conclusively.
149
5.7 Smothered Mate
The smothered (mate) theme involves checkmating the enemy king using a knight such
that all of the king’s flight squares are blocked by its own pieces, (defended) opponent
pieces, or a combination of both (Hooper and Whyld, 1996). This is sometimes simply
referred to as the ‘smothered’ theme; in such cases, it does not necessarily involve
checkmating the opponent; other pieces can also be ‘smothered’.
The smothered mate theme is similar to the ‘self-block’ theme (more common to
composers), which is not specific about the checkmate itself or the piece delivering it,
except that the king is blockaded by his own pieces (Howard, 1967). For the purpose of
this research and to cover a wider thematic scope, the smothered mate was not restricted
to using the knight. The smothered mate can happen at any point in the game and is far
more common in the corners and edges of the board than the centre because there are
fewer flight squares. Figure 5.11 illustrates an example of this theme. The proposed
formalization is shown in equation 5.7.
XABCDEFGHY
8ktr-+-+-+(
7zppsN-+-+-'
6-+-+-+-+&
5+-+K+-+-%
4-+-+-+-+$
3+-+-+-+-#
2-+-+-+-+"
1+-+-+-+-!
xabcdefghy
Figure 5.11 The Smothered Mate
150
( )17 .c pT s r s−= ∑ (5.7)
sp = smothering pieces (those in the field of the enemy king)
sc = the smothered constant (i.e. 101)
The constant is derived from the ideal smothered mate at the centre of the board with
the maximum of eight pieces around the king. Based on the original piece set, the piece
power of the most powerful 8 pieces in descending order are the queen (27), 2 rooks
(14+14), 2 bishops (13+13), 2 knights (8+8) and a pawn (4) for a total of 101. The piece
power is used instead of the piece value because it is less likely that the more powerful
pieces should crowd around the opponent king and act primarily as blockades.
They are usually put to better use such as controlling more space on the board and
attempting to checkmate the opponent king. The most common form of this theme is at
the corner of the board with a rook and two pawns. Simply using the number of pieces
in the enemy king’s field does not account for the variety of pieces on the board. No
differentiation was made between the colours of the pieces blockading the king. There
were also no liabilities of consequence to this theme. Either the checkmate is true on
account of it or it is not.
5.8 Cross-check
The cross-check occurs when a player responds to a check with a reciprocal check. It is
notably a chess theme where manoeuvres by both players are rightfully taken into
account for aesthetic purposes. The cross-check is achieved by moving the king out of
harm’s way to uncover a discovered check on the opponent’s king or intervening with a
piece that simultaneously gives check. It does not usually include a check that results
from capturing the checking piece (Hooper and Whyld, 1996; Rice, 1997). This, rules
151
out common positions where a series of checks is merely the result of repeated
exchanges on the same square. Equation 5.8 shows how the aesthetic score for the
theme is calculated.
( ) ( )( )118
1. , . , 2n
c n k nT c n d cp o r cp n−− = + ≥ ∑ (5.8)
n = number of consecutive checks, cc = crosscheck constant (i.e. ∑ plies)
cpn = crosscheck piece, ok = opponent king
The cross-check constant is equivalent to the number of plies in the combination. A
cross-check pair could occur in any two consecutive plies in the combination. Two
cross-check pairs could also be separated by a non-check ply (n = 4). A cross-check pair
and single check in the same combination, if separated by a non-check move counts
only as a single cross-check pair (n = 2). Only if they occur consecutively do they count
as three checks (n = 3). White not need have the first check. The following composition
shows how the cross-check is calculated.
XABCDEFGHY
8-+-+-+-+(
7+-+p+-+-'
6R+-sN-+r+&
5mK-+-+-vL-%
4-+-+p+-+$
3mkp+-+p+-#
2-tr-snl+-+"
1+nsN-vlq+-!
xabcdefghy
William M Greenwood, Cassell's Family Paper, 1857
Figure 5.12 Aesthetic Assessment of the Cross-check
152
In Figure 5.12, the moves are: 1. Be7 Nc4+ 2. Kb5+ Na5+ 3. Nc4#. There are four
consecutive checks. The score is calculated as shown below.
( )1 1 1 1 1- 5 . 4 2 8 3 14 3 13 4 13 1.001cross check − − − − − = + × + × + × + × =
Note that on the last move, the checkmating piece is not the knight - even though it was
the one that moved - but the bishop on e7. This is because technically, the knight is
pinned to its king by the bishop on e2. On Black’s first move, his knight is also the
moving piece but here it is not pinned so the bishop on e1 is not considered the
checking piece. The other moves do not have double checks.
5.9 Promotion
Pawn promotion occurs when a pawn reaches the end of the board and promotes either
to a queen, rook, bishop or knight. The most common choice is the queen but promotion
to a knight is not uncommon, especially when it is prudent to do so. In the 15th century,
some argued that promotion resulting in a plurality of queens was tantamount to
condoning adultery. Others said that it should not be possible to have more power than
the initial piece set (Hooper and Whyld, 1996). There are even cases where promotion
to a bishop or rook is necessary (e.g. where promoting to a queen gives stalemate). One
of the best examples of the latter is the Saavedra position from the late 19th century
(Berry, 2005). The following diagram (Figure 5.13) illustrates it. The Saavedra position
is named after the Reverend Saavedra who in the late 19th century discovered a win in a
position thought to have been drawn.
153
XABCDEFGHY
8-+-+-+-+(
7+-+-+-+-'
6-mKP+-+-+&
5+-+r+-+-%
4-+-+-+-+$
3+-+-+-+-#
2-+-+-+-+"
1mk-+-+-+-!
xabcdefghy
White to play and win
Figure 5.13 The Saavedra Position
The solution to this endgame study is as follows:
1. c7 Rd6+
2. Kb5 (2. Kc5 Rd1 and 3. ... Rc1) Rd5+
3. Kb4 Rd4+
4. Kb3 Rd3+
5. Kc2 Rd4
6. c8=R (6. c8=Q Rc4+ 7. Qxc4 stalemate) Ra4
7. Kb3
At this point, Black loses either the rook or the game. The solution, however, is not
perfect as computer analysis (using endgame tablebases) has shown that Black can
delay checkmate (by 3 moves) by playing 3. … Kb2 instead (Lokasoft, 2008). This
reiterates the fact that aesthetically, an entirely ‘correct’ solution is not necessarily a
prerequisite (see subsection 3.3.3). Underpromotion is considered more beautiful
because it is both paradoxical and economical. The formalization is therefore given as
shown in equation 5.9.
154
( ) 19 .c pT p v p −= (5.9)
pc = promotion constant (i.e. 3), pp = promotion piece
Promotion to a queen results in the lowest score whereas promotion to a knight or
bishop, the maximum score of 1. Promotion and underpromotion are common in
problem composition but they also occur in real games, especially in the endgame when
there are fewer pieces to impede the march of passed pawns.
5.10 Switchback
The switchback is the return of a single piece to its initial square, either immediately or
later in the move sequence (Levitt and Friedgood, 2008). For the purpose of this
research, it also includes the similar ‘rundlauf’ or round-trip theme where a piece leaves
a square, and then later in the solution returns to it by a circuitous route (e.g. a rook
moves e3-g3-g5-e5-e3) whereas in the switchback, a piece leaves a square and then later
in the solution returns to it by the same route (e.g. a rook moves e3-e5-e3). Only pawns
are incapable of such a manoeuvre because they can only move forward. Given the
scope of analysis, this theme can occur at most twice in the move sequence.
The author at first thought the switchback could occur only once given a mate-in-3
combination, but after programming the detection algorithm and some testing,
CHESTHETICA (see Appendix D) revealed combinations in which the theme occurred
technically twice. If from the initial position a piece moves to another location on move
1, then back to its original square on move 2 (this counts as one switchback) and back
again to said location on move 3 (another switchback), both squares are technically
155
revisited. The oversight was simply that the original square of the moving piece also
counts.
Distance and piece power in relation to the move played do not apply to the switchback.
The distance to piece power ratio used in the other theme formalizations (e.g. the fork,
see section 5.1) apply only where there are things like attacker and target. There are no
‘attackers’ or ‘targets’ in the switchback. The aesthetic score for this theme is therefore
simply equal to the number of times it is detected divided by its benchmark i.e. 2 (see
equation 5.10).
110 . cT n sb −= (5.10)
n = number of switchbacks, sbc = switchback constant (i.e. 2)
5.11 Points of Evaluation for the Themes
Table 5.2 lists the ‘points of evaluation’ in a combination (see section 3.8) for each
theme explained in this chapter, given the scope of analysis (see section 3.7). These are
the points or stages in a mate-in-3 combination where each theme is computationally
evaluated using its proposed formalization. Not all the themes are evaluated after every
move. The zugzwang is evaluated only once after the second move given the constraints
explained in section 5.6. The smothered mate is evaluated in the final position for
consistency with the role of the other pieces in the theme. The cross-check and
switchback themes each require more than one move to demonstrate so their results are
calculated only in the final position. The actual points of evaluation for these two
themes are given in brackets anyway.
156
Table 5.2 Points of Evaluation for the Themes
Theme Point(s) of Evaluation 1 Fork 1, 2, 3 2 Pin 1, 2, 3 3 Skewer 1, 2, 3 4 X-Ray 1, 2, 3 5 Discovered/Double Attack 1, 2, 3 6 Zugzwang 2 7 Smothered Mate 4 8 Cross-check 4 (1.5, 2, 2.5, 3) 9 Promotion 1, 2, 3 10 Switchback 4 (2, 3)
5.12 Chapter Summary
In this chapter, the theme formalizations developed for this research were explained in
detail. These included all the ten selected themes (see section 3.4) which are also listed
in Table 5.2. The first five themes have relatively complex formalizations given the
many parameters involved. Examples of how to calculate the aesthetic scores for these
themes based on their formalizations were provided where appropriate. The
formalizations for the following five themes are comparatively less complex. However,
this does not imply that they are any less effective. From the explanations provided in
their respective sections, it can be seen that the relevant factors pertaining to those
themes were sufficiently taken into account in the design of their evaluation functions.
Themes are essentially evaluated after every move in a combination except where they
might require more than one move to demonstrate or pertain only to the checkmate. In
those cases, they are evaluated at the end.
It is important to note that the aesthetic evaluation of a single theme in a particular move
does not necessarily capture the overall beauty of the combination or ensuing position
157
which observers tend to see all at once. The evaluation of a single instance of a theme is
just one aesthetic component of many that might be present. The other features of the
position, therefore, are not neglected but merely assessed separately. It is also
noteworthy that a featured theme (e.g. in a composition) which is not specifically
accounted for, is not necessarily neglected completely in terms of cumulative aesthetic
assessment. It will certainly play into (at least) the calculations of the aesthetic
principles to some degree (see chapter 4). The next chapter explains the experiments
that were performed to validate the proposed model. Analyses and discussions of the
results are also included.
158
CHAPTER 6: EXPERIMENTAL RESULTS and DISCUSSIONS
6.0 The Six Experiments Performed
This chapter explains the design of six experiments performed to validate the proposed
model of aesthetics. The results of each experiment are presented, analyzed and
discussed in their respective sections. Where appropriate, the discussion is presented in
a subsection. Experiment 1 (section 6.1) tests if the selected aesthetic principles and
themes are common to both domains (i.e. compositions and real games), consistent with
the proposed conceptual framework for aesthetics presented in section 3.2. Experiment
2 (section 6.2) compares combinations between and within domains in terms of their
cumulative aesthetic principle scores using the formalizations developed for them (see
chapter 4).
Experiment 3 (section 6.3) compares the scores with regard to themes (see chapter 5).
Experiment 4 (section 6.4) compares the cumulative aesthetic scores (seven principles +
ten themes). Experiment 5 (section 6.5) tests the computational aesthetic assessment for
conformity with authoritative human assessment in chess literature. Experiment 6
(section 6.6) tests for positive correlation with human chess player aesthetic assessment.
The results are based on four surveys that were conducted with members and visitors to
an online chess community.
Data Sets and Constraints
A total of 19,344 direct mate-in-3 compositions and 12,552 mate-in-3 combinations
from tournament games between expert players managed to be obtained as data sets for
159
these experiments (Meson Chess Problem Database, 2008; Mega Database, 2008).
These compositions and combinations had no particular order to them (and were
effectively random). Tournament games fall under the category of ‘real’ or ‘over-the-
board’ games which are distinct from compositions. Games between expert players (i.e.
Elo ≥ 2000) were used to maximize aesthetics and effectiveness in that domain, and
minimize bias against them (as might result from the unsound play of amateurs).
Introducing additional criteria (e.g. games before or after 1970) would have severely
limited the number of combinations to analyze. As far as possible, the main line or
variation to a composition was used but in cases where it was unknown, one line was
selected at random.
Compositions may feature side variations but given that real games do not, only one line
(i.e. the main one) could be used. The ‘aesthetic content’ of a composition is not
drastically compromised as a result because variations are seldom critical to the solution
of direct-mate problems (Hooper and Whyld, 1996). Additionally, aesthetic evaluation
in this thesis pertains to it as a separate component not necessarily linked to composition
conventions or brilliancy characteristics in real games (see section 3.2). The cumulative
aesthetic scores were rounded to three decimal places. A multiplier (e.g. by 1,000) to
make them whole numbers was not used because this could be misleading in terms of
how reliably precise those values are when compared to say, human assessment (see
subsection 6.6.5). The author also wanted to preserve, for potential direct comparative
purposes, the purity of the aesthetic evaluation scores in relation to the metrics used.
160
6.1 Experiment 1: Frequencies
The first experiment performed compared two sets of 9,672 direct mate-in-3
compositions and two sets of 6,276 mate-in-3 combinations from tournament games
between expert players. They were randomly selected from the data set of 19,344
compositions and 12,552 games (see previous section) using a random function
programmed into CHESTHETICA (see Appendix D). The experiment intended to
establish whether or not the chosen aesthetic principles and themes were a suitable
‘common ground’ for aesthetic evaluation between both domains in line with the
conceptual framework for aesthetics proposed in section 3.2.
This does not mean that the frequency of occurrence for each aesthetic principle or
theme should be similar in both domains (which is unlikely). It only means that there
should not be a statistically significant difference between domains in terms of the
average frequency of the aesthetic principles and the average frequency of the themes.
The average frequencies or their differences themselves are not presented in the text to
keep the discussion relevant (but can be derived from the numerical data in the charts);
however, presented are the statistical results of significance testing on those differences,
such as the t and P values. The combinations were tested for all the seven aesthetic
principles (see chapter 4) and ten themes (see chapter 5). For this experiment, results
pertaining to the aesthetic principles are explained after the themes since there is more
to say about the latter.
161
6.1.1 Frequencies of the Themes
Figure 6.1 shows the frequency of the ten themes (see chapter 5) in the combinations.
The frequency of a theme was calculated as the number of times it was detected in the
moves of the combinations divided by the number of times it was tested for detection.
For each combination, the fork, pin, skewer, x-ray, ‘discovered/double attack’ and
‘promotion’ themes were tested for, three times (after every move by White). The
‘switchback’ was tested for, twice; after White’s second and third move. The zugzwang
tested for, once; after Black’s reply to White’s second move (i.e. after ply 4). The
‘smothered mate’ was tested for, once; in the final position; and the ‘cross-check’ tested
for, four times (first after two ply, and then after each of the remaining three ply). These
are equivalent to the points of evaluation for themes (see section 5.11) but explained
here in a way relevant to the experiment.
Figure 6.1 Frequencies of Themes in the Combinations
Fork Pin Skewr X-ray DDA Zugzw Smoth Cross Prom SwitchCOMP Set 1 6.07 0.82 1.32 0.06 6.79 15.39 0.14 0.87 1.54 1.25COMP Set 2 6.31 0.83 1.31 0.04 6.79 15.32 0.14 0.90 1.64 1.23TG Set 1 5.42 1.77 1.96 0.04 2.93 0.81 0.86 0.04 1.23 1.45TG Set 2 4.94 1.73 2.11 0.01 2.64 0.70 0.91 0.04 1.14 1.53
0
2
4
6
8
10
12
14
16
Freq
uenc
y (%
)
Themes
COMP Set 1
COMP Set 2
TG Set 1
TG Set 2
162
At least one instance of any of these themes occurred in 10,132 out of the 19,344
composition combinations or 52.38% of the time. Only the formalizations for two
themes (i.e. the pin and discovered/double attack) were observed as capable of resulting
in a negative evaluation score in actual composition and tournament game
combinations. A negative cumulative theme score rarely occurs (only 0.168% and
0.045% of the time for each domain, respectively) and rarer still in such a way that
results in exactly 0. Therefore, in cases where the themes in a combination cumulatively
scored precisely 0, it was safely assumed that none occurred.
For the tournament games, at least one instance of a theme occurred in 4,415 out of
12,552 combinations or 35.17% of the time. Compositions in total had a higher
frequency in 6 of the 10 themes. A higher frequency in all of the themes (for either
group) would tend to imply that the selection of themes was biased. A slightly greater
majority for compositions was nonetheless expected (it follows composition
convention, see subsection 3.3.1). An even lower frequency of these themes would be
expected in less intense (i.e. not checkmate) stages of the game. Note, however, that a
higher frequency of themes in a combination does not necessarily imply a higher
aesthetics score for it. The formalizations developed for the themes are dynamic and
often result in different aesthetic scores from one instance of a theme to another.
There was no statistically significant difference (TTUV, 2T, SL 5%) between the
average frequencies of these themes for both sets of compositions (see Abbreviations);
t(18) = -0.012, P=0.991. The same was true between the tournament game sets; t(18) =
0.112, P=0.912. There was also no statistically significant difference between the
average frequency of themes for compositions and tournament games in total; t(11) =
1.143, P=0.277. This further supports the idea that they should not be exclusive to either
163
group (see section 3.4) even though the zugzwang seems to appear much more
frequently in compositions. The zugzwang typically requires more planning than the
other themes and seldom occurs by chance in real games, except perhaps in the
endgame (which is why it was still included). The zugzwang is, in fact, a legitimate
tactic employed in real games (Averbakh, 1992; Königsberg, 2007).
6.1.2 Frequencies of the Aesthetic Principles
Only three of the seven aesthetic principles (see chapter 4) are capable of exhibiting
variations in frequency. These include ‘violate heuristics successfully’, ‘winning with
less material’ and ‘sacrifice material’. The rest, by definition, occur consistently in both
domains regardless. For example, ‘use the weakest piece possible’ (see section 4.2)
applies to the checkmating piece and will always occur in a mating combination.
Therefore, frequency testing involved only the three principles mentioned.
Figure 6.2 Frequencies of Aesthetic Principles in the Combinations
VH WWLM SACCOMP Set 1 80.18 15.21 47.55COMP Set 2 80.44 14.68 46.56TG Set 1 52.92 29.08 22.10TG Set 2 53.51 27.66 22.10
0102030405060708090
100
Freq
uenc
y (%
)
Aesthetic Principles
COMP Set 1COMP Set 2TG Set 1TG Set 2
164
Figure 6.2 shows the results. There was no difference of statistical significance (TTUV,
2T, SL 5%) between the average frequencies of these aesthetic principle for the
composition sets; t(4) = 0.016, P=0.988; tournament game set; t(4) = 0.021, P=0.985; or
in total; t(3) = 0.61, P=0.585.
6.1.3 Discussion
Given that the average frequencies of the aesthetic principles and themes showed no
significant difference between the domains of compositions and real games, they were
validated as a suitable ‘common ground’ to evaluate aesthetics between said domains, in
line with the proposed conceptual framework for aesthetics (see section 3.2). Even
though these aesthetic principles and themes were selected for suitability based on chess
literature (see section 3.4), experimental validation is still recommended.
It is quite possible that a different selection of aesthetic principles and, especially,
themes (unlike those featured here) might also have mean frequencies that are not
significantly different between the domains. However, suitability based on chess
literature would then still be recommended. For example, if the literature states that a
particular theme or principle is exclusive to the domain of compositions, it would
probably be prudent to exclude it. In any case, something would be wrong with the
selection of say, themes if a statistically significant difference was detected between its
mean frequencies in both domains. Rather than replacing the whole set, perhaps
replacing just one or two suspected ‘domain specific’ themes would be more efficient.
165
6.2 Experiment 2: Evaluation of the Aesthetic Principles
The second experiment investigated the computational evaluations for the aesthetic
principles. These evaluations were based on the formalizations developed for said
principles (see chapter 4). The intention of performing this experiment was to see if
there was any difference between the evaluation scores of the aesthetic principles for
compositions and combinations in real games. Since the board, pieces, rules, move
length and objective (i.e. checkmate) are the same in all the sets, it stands to reason that
any significant difference between sets is aesthetic in nature. It also stands to reason that
within the same domain, there should not be a statistically significant difference in the
evaluation scores for each principle.
A difference is expected between domains because one has the benefit of a composer
while the other does not. In short, this experiment attempts to validate the individual
formalizations for the aesthetic principles in terms of their ability to capture them. Table
6.1 shows the average scores for the aesthetic principles. Their names have been
shortened to save space (see ‘Abbreviations’).
Table 6.1 Average Scores for Aesthetic Principles
Aesthetic
Principles TG COMP Total
Set 1 Set 2 Set 1 Set 2 TG COMP 1 VH 0.071 0.068 0.227 0.226 0.070 0.227 2 WPP 0.255 0.259 0.313 0.314 0.257 0.314 3 APP 0.591 0.597 0.652 0.655 0.594 0.654 4 WWLM 0.067 0.068 0.129 0.136 0.068 0.132 5 Economy 0.083 0.089 0.294 0.294 0.086 0.294 6 Sac. Mat. 0.212 0.213 0.291 0.293 0.212 0.292 7 Sparsity 0.381 0.382 0.395 0.396 0.381 0.395
166
On average, the compositions scored higher than the tournament games in all of the
seven aesthetic principles. The greatest disparity was in the principle of ‘economy’
whereas the smallest was in terms of sparsity. Within domains the differences were
generally smaller. The significance of these differences can be seen in Table 6.3.
Table 6.2 Standard Deviations of Average Aesthetic Principle Scores
Aesthetic
Principles TG COMP Total
Set 1 Set 2 Set 1 Set 2 TG COMP 1 VH 0.081 0.074 0.190 0.190 0.077 0.190 2 WPP 0.174 0.181 0.202 0.202 0.177 0.202 3 APP 0.244 0.243 0.232 0.232 0.243 0.232 4 WWLM 0.050 0.051 0.102 0.106 0.050 0.104 5 Economy 0.179 0.184 0.197 0.200 0.181 0.199 6 Sac. Mat. 0.153 0.156 0.206 0.208 0.154 0.207 7 Sparsity 0.098 0.102 0.135 0.132 0.100 0.134
In total (see Table 6.2), compositions showed a greater standard deviation than
tournament games in all the sets for all the aesthetic principles except the third one (i.e.
use all of the piece’s power). In many fields, standard deviations simply tend to
correlate with means.
Table 6.3 shows the significance of the differences in means between aesthetic principle
scores for each aesthetic principle in each set pair of each domain and cumulatively.
The total number of occurrences of the aesthetic principle in each set is provided below
the significance reading. S = significant, NS = not significant, S1=Set 1 and S2=Set 2.
All were tested using: TTUV, 2T, SL 1%. No assumption was made about which
domain would score higher aesthetically; only that there would be a difference. The
only significant differences were found between the domains of compositions and
tournament games. Within either domain, the differences in means were not significant.
167
Table 6.3 Significance of Differences between Mean Aesthetic Principle Scores
Aesthetic
Principles Set 1 vs. Set 2 (TG vs. TG)
Set 1 vs. Set 2 (COMP vs. COMP)
Set 1+2 vs. Set 1+2 (COMP vs. TG)
1
VH
NS t(6610)=1.798,
P=0.072 (S1=3321, S2=3358)
NS t(15533)=0.336,
P=0.737 (S1=7755, S2=7780)
S t(22156)=87.478,
P<0.01 (C=15537, TG=6459)
2
WPP
NS t(12533)=-1.19,
P=0.234 (S1=6276, S2=6276)
NS t(19342)=-0.631,
P=0.528 (S1=6276, S2=6276)
S t(29167)=26.25,
P<0.01 (C=19344, TG=12552)
3
APP
NS t(12550)=-1.363,
P=0.173 (S1=6276, S2=6276)
NS t(19342)=-0.779,
P=0.436 (S1=6276, S2=6276)
S t(25885)=21.82,
P<0.01 (C=19344, TG=12552)
4
WWLM
NS t(3538)=-0.563,
P=0.574 (S1=1825, S2=1736)
NS t(2874)=-1.72,
P=0.086 (S1=1471, S2=1420)
S t(3988)=30.7,
P<0.01 (C=2891, TG=3561)
5
Economy
NS t(12541)=-1.927,
P=0.054 (S1=6276, S2=6276)
NS t(19337)=-0.143,
P=0.886 (S1=6276, S2=6276)
S t(28543)=96.167,
P<0.01 (C=19344, TG=12552)
6
Sac. Mat.
NS t(2771)=-0.28,
P=0.779 (S1=1387, S2=1387)
NS t(9092)=-0.567,
P=0.571 (S1=4599, S2=4503)
S t(6087)=21.85,
P<0.01 (C=9102, TG=2774)
7
Sparsity
NS t(12521)=-0.474,
P=0.636 (S1=6276, S2=6276)
NS t(19334)=-0.376,
P=0.707 (S1=6276, S2=6276)
S t(31246)=10.85,
P<0.01 (C=19344, TG=12552)
The formalizations are therefore considered validated in terms of being able to capture
aesthetics in the game with regard to these aesthetic principles. Cumulative aesthetic
principle assessment is tested in Experiment 4 (section 6.4).
6.3 Experiment 3: Evaluation of the Themes
The third experiment is in principle the same as Experiment 2 except that it was
intended to validate the theme formalizations developed. Table 6.4 shows the average
168
scores for the themes. Enemy pawns were excluded as viable targets for the fork, pin,
skewer, and ‘discovered/double attack’ themes (see chapter 5).
Table 6.4 Average Scores for the Themes
Themes TG COMP Total Set 1 Set 2 Set 1 Set 2 TG COMP
1 Fork 0.462 0.461 0.446 0.448 0.462 0.447 2 Pin 0.581 0.575 0.631 0.651 0.578 0.640 3 Skewer 0.776 0.788 0.693 0.685 0.782 0.688 4 X-ray 0.424 0.537 0.465 0.525 0.446 0.492 5 DDA 0.650 0.647 0.598 0.603 0.649 0.600 6 Zugzwang 0.047 0.047 0.110 0.106 0.047 0.108 7 Smothered 0.374 0.392 0.357 0.368 0.383 0.363 8 Cross-check 0.480 0.489 0.532 0.530 0.484 0.531 9 Promotion 0.389 0.367 0.608 0.639 0.378 0.624 10 Switchback 0.500 0.500 0.502 0.513 0.500 0.507
The fork, skewer, ‘discovered/double attack’ and ‘smothered mate’ scored on average
higher in tournament games than in compositions. The other six themes scored lower in
tournament games. In principle, a particular theme can be executed more effectively and
aesthetically in a real game than in a composition. This is especially true when a theme
is not necessarily the ‘main idea’ of a composition, as is probably the case here. In a
composition, the theme that forms the ‘main idea’ is likely to possess a unique or
distinguished configuration compared to those that are secondary or coincidental.
Nevertheless, themes of the latter variety usually still bear the mark of composition
because composers cannot be unaware of them (Albrecht, 2000). The ‘main idea’
should not, however, be conflated with the ‘main line’ (see section 6.0) which is simply
the primary solution to a chess problem.
A theme that forms the ‘main idea’ may not necessarily be among those detected here.
For example, it could be an exotic theme such as the Pickaninny or Grimshaw which is
quite specific to compositions. In a ‘common ground’ of themes, neither domain is
169
necessarily considered more beautiful. Even award-winning compositions may not be
the most beautiful let alone for a particular, common theme (Hochberg, 2005).
Nevertheless, there should be an aesthetic difference in the way they are configured
because one is composed while the other is not.
As with the aesthetic principles (see section 6.2), compositions tended to have a greater
standard deviation than real games for themes as well (see Table 6.5). Exceptions
include the ‘discovered/double attack’ and ‘smothered mate’ which were higher for real
games. The pin and x-ray were quite close. For the pin, sets 1 and 2 of the tournament
games had a greater standard deviation than set 2 of the compositions, but not in total.
The x-ray had a greater standard deviation for set 1 of the tournament games than set 2
of the compositions, but again not in total.
Table 6.5 Standard Deviations of Average Scores for Themes
Themes TG COMP Total
Set 1 Set 2 Set 1 Set 2 TG COMP 1 Fork 0.077 0.077 0.078 0.085 0.077 0.082 2 Pin 0.153 0.144 0.160 0.139 0.148 0.150 3 Skewer 0.143 0.141 0.155 0.148 0.142 0.154 4 X-ray 0.144 0.110 0.156 0.125 0.141 0.144 5 DDA 0.291 0.299 0.267 0.259 0.295 0.263 6 Zugzwang 0.050 0.053 0.133 0.126 0.051 0.129 7 Smothered 0.161 0.139 0.115 0.106 0.150 0.109 8 Cross-check 0.027 0.022 0.123 0.113 0.024 0.118 9 Promotion 0.164 0.129 0.314 0.319 0.149 0.317 10 Switchback 0.000 0.000 0.032 0.079 0.000 0.060
Table 6.6 shows the significance of the differences in means for each theme in each set
pair of each domain and in total. The total number of occurrences for each theme in
each set is provided below the significance reading. S = significant, NS = not
significant, S1=Set 1 and S2=Set 2. All were tested using: TTUV, 2T, SL 1%. As with
170
Experiment 2, there was no assumption about which domain would score higher
aesthetically; only that there would be a difference.
Table 6.6 Significance of Mean Differences of Theme Scores
Themes Set 1 vs. Set 2 (TG vs. TG)
Set 1 vs. Set 2 (COMP vs. COMP)
Set 1+2 vs. Set 1+2 (COMP vs. TG)
1
Fork
NS t(1934)=0.209,
P=0.834 (S1=1021, S2=931)
NS t(3580)=-0.828,
P=0.408 (S1=1761, S2=1830)
S t(4210)=-6.572,
P<0.01 (C=3591, TG=1952)
2
Pin
NS t(657)=0.578,
P=0.564 (S1=334, S2=326)
NS t(465)=-1.466,
P=0.143 (S1=237, S2=242)
S t(1022)=-6.92,
P<0.01 (C=479, TG=660)
3
Skewer
NS t(759)=-1.199,
P=0.231 (S1=369, S2=398)
NS t(761)=0.729,
P=0.466 (S1=384, S2=380)
S t(1518)=-12.47,
P<0.01 (C=764, TG=767)
4
X-ray
NS t(2)=-1.217,
P=0.348 (S1=8, S2=2)
NS t(27)=-1.144,
P=0.263 (S1=16, S2=13)
NS t(16)=0.88,
P=0.392 (C=29, TG=10)
5
DDA
NS t(1029)=0.157,
P=0.876 (S1=551, S2=497)
NS t(3936)=-0.587,
P=0.557 (S1=1971, S2=1971)
S t(1519)=-4.822,
P<0.01 (C=3942, TG=1048)
6
Zugzwang
NS t(89)=-0.062,
P=0.95 (S1=51, S2=44)
NS t(2962)=0.977,
P=0.329 (S1=1489, S2=1482)
S t(136)=10.56,
P<0.01 (C=2971, TG=95)
7
Smothered
NS t(105)=-0.63,
P=0.53 (S1=54, S2=57)
NS t(26)=-0.252,
P=0.803 (S1=14, S2=14)
NS t(56)=-0.816,
P=0.418 (C=28, TG=111)
8
Cross-check
NS t(18)=-0.855,
P=0.404 (S1=10, S2=10)
NS t(674)=0.186,
P=0.852 (S1=337, S2=348)
S t(53)=6.61,
P<0.01 (C=685, TG=20)
9
Promotion
NS t(432)=1.57,
P=0.117 (S1=231, S2=215)
NS t(920)=-1.494,
P=0.136 (S1=447, S2=477)
S t(1367)=19.5,
P<0.01 (C=924, TG=446)
10
Switchback
NS Means
Identical (S1=182, S2=192)
NS t(312)=-1.918,
P=0.056 (S1=241, S2=237)
S t(477)=2.663,
P<0.01 (C=478, TG=374)
171
There was no statistically significant difference between the mean aesthetic scores of
individual themes found in the two sets of tournament games suggesting that themes
there come about in a similar format. The same was true when comparing compositions.
While significant differences between compositions sets (for certain themes) are
perhaps more likely - more creativity is typically involved than in tournament games -
the results reflect the ideal scenario because the compositions were not selected with
these themes in mind and are therefore generic in that sense. In comparing compositions
and tournament games (in total), however, there was a significant difference between
the means of all the themes except for the x-ray and ‘smothered mate’.
There are perhaps three possible explanations for this. First, the evaluation functions
proposed for them may not have been designed in a way that captured their aesthetic
content. This cannot be ruled out. Second, the relatively low number of occurrences in
both domains for the x-ray and in compositions for the ‘smothered mate’ might have
made statistical analysis unreliable. However, if that were the case, then the results for
the ‘cross-check’ theme might also be questionable. Third, and this is most likely in the
author’s opinion, is that certain themes may not, in fact, display aesthetic differences
between domains as prominently as other themes, if at all.
Complexity (of the formalization) could also be a factor. If more properties are
evaluated – possibly resulting in a wider possibility of scores - it might better capture
the aesthetics of themes like the ‘smothered mate’ and x-ray. On the downside,
increased complexity could also result in diminishing returns from a computational
standpoint. A more complex version of the smothered mate formalization that took into
account the colour of the piece in the king’s field was also tested but there was no
difference in the results. There was actually no necessity or basis in the literature
172
surveyed to include this piece. The original formalization was designed in a way that
accounted for all that was considered pertinent to the theme.
A simpler version of the x-ray formalization and an equally complex alternative did not
improve its results either. They are not presented here so as to avoid confusion with the
original formalizations proposed. Note that altering an aesthetics formalization, once it
has been satisfactorily and consistently constructed (e.g. based on a particular
development methodology), is not recommended in this kind of experimentation. They
should be developed beforehand and their performance reported as is.
The theme constants and other variables were not arbitrarily adjusted because the
fundamental idea behind the approach taken in this research was to base assessment on
inherent metrics in the game rather than on taste, personal experience or tuned values
(to a particular data set). The advantage of these metrics (see subsection 3.5.2) is that
they are generic enough to describe aspects of the game in a wide variety of
circumstances, making them ideal for aesthetic analysis which is difficult to
demonstrate in crisp terms (e.g. comparing one position to another and proving that
either is more beautiful). These metrics are also (for the most part) universally accepted
among chess players (see subsection 3.5.1). In summary, this experiment suggests that
all the theme formalizations proposed - with the possible exceptions of the ones for the
x-ray and ‘smothered mate’ - are able to capture aesthetic aspects of their respective
themes to a reasonable degree.
173
6.4 Experiment 4: Cumulative Evaluation
Using the same data sets as the previous experiments, cumulative aesthetic scores for
the combinations based on all the seven aesthetic principles and ten themes were
computed. The experiment was designed to test if compositions scored higher
aesthetically (in total) than real games and if that difference was statistically significant.
Despite the lack of a significant difference between domains (see previous section), the
x-ray and ‘smothered mate’ themes were still included here because at worst, their
assessments would be inconsequential to the domains. Additionally, the overall
aesthetic value of a combination - be it in a real game or composition - would be better
represented not excluding these themes, if detected. Table 6.7 shows the mean
cumulative aesthetic scores and their standard deviations.
Table 6.7 Average Cumulative Aesthetic Scores for the Combinations
TG COMP Total
Set 1 Set 2 Set 1 Set 2 TG COMP 1.658
SD 0.618 1.663
SD 0.614 2.317
SD 0.719 2.329
SD 0.732 1.660
SD 0.616 2.323
SD 0.725
Compositions, as would be expected, scored cumulatively higher aesthetically than
tournament games. The difference of 0.663 points between domains (in total) was
statistically significant (TTUV, 2T, SL 1%); t(29702) = 87.423, P<0.01. This represents
a score increase of approximately 40% compared to real games. The differences within
each domain, however, were much smaller and not statistically significant, suggesting
their aesthetic content is similar in value. Between tournament game sets, the difference
was only 0.005; t(12549) = -0.475, P = 0.63 and between composition sets it was only
0.012; t(19336) = -1.146, P = 0.252. Figure 6.3 shows a chart depicting the cumulative
174
aesthetic scores of both domains in total. They have been arranged in ascending order
for illustrative purposes.
Figure 6.3 Cumulative Aesthetic Scores for Combinations
The highest score for a composition from the data sets was 6.674 and for a tournament
game combination it was 5.239. The former featured all seven aesthetic principles and
five instances of the themes. The lowest scoring were 0.613 and 0.444, respectively. The
positions are shown in Figures 6.4 and 6.5.
0
1
2
3
4
5
6
7
8
1 1501 3001 4501 6001 7501 9001 105011200113501150011650118001
Scor
e
Positions
Compositions vs. Tournament Games
COMPTG
175
XABCDEFGHY
8-+-+-vL-+(
7+K+-tR-+l'
6-+-mk-+-+&
5+P+-+-tR-%
4-+-trp+-+$
3+-sN-+-+-#
2-+n+-vl-sn"
1+-+-+-+-!
xabcdefghy
XABCDEFGHY
8r+l+-tr-mk(
7+pzp-snptRp'
6-+-sn-+-+&
5zp-+-+-+q%
4-+L+-+-+$
3+-vL-+-+-#
2P+-+-zPPzP"
1tR-+-+-mK-!
xabcdefghy (a) Sam Loyd, British Chess Magazine,
1910 (score: 6.674) (b) Voight, M. vs. Parindra, A.,
Hamburg-ch int, 2004 (score: 5.239) 1. Nd5 Kc5 2. Rc7+ Kxb5 3. Nc3# 1. Rxf7+ Kg8 2. Rg7+ Kh8 3. Rg8#
Figure 6.4 Highest Scoring Combinations (a) COMP, (b) TG
As an example of actual calculations, the assessment for the combination in Figure
6.4(a) is shown below. See chapters 4 and 5 for an explanation of these principles and
themes. See section 3.8 for information about the POE (point of evaluation).
Initial Position (POE = 0):
Win with less material = [ ]138 . 18 17− − = 0.026
Sparsity = ( ) 1113 .10 1−
− + = 0.565
After 1. Nd5 (POE = 1):
Violate Heuristics = capture enemy material (bishop, h7) + increase piece mobility
= (0.333 + 37-1.(37-35)).4-1 = 0.097
All of the Piece’s Power (knight) = 2.8-1 = 0.25
176
After 2. Rc7+ (POE = 2):
All of the Piece’s Power (rook) = 2.14-1 = 0.143
Fork = undefended bishop (h7) and king (c5) = ( )( )1 1 137 . 13 2 5.14 2.14 0− − − + + + −
= 0.419
Skewer = king (c5) and knight (c2) = ( ) ( )1 1 119 . 10 3 5.14 0.8 0 0− − −+ + − + + = 0.703
Double check = rook (c7) and bishop (f8) = ( ) ( )1 1 120 . 10 10 2.14 3.13 0 0− − − + + + − +
= 1.019
After 3. Nc3# (POE = 3):
Weakest Piece Possible to Checkmate (knight) = 4.8-1 = 0.5
All of the Piece’s Power (knight) = 2.8-1 = 0.25
Double check (mate) = rook (g5) and knight (c3)
= ( ) ( )1 1 120 . 10 10 2.8 5.14 0 0− − − + + + − + = 1.03
Final Position (POE = 4):
All of the Piece’s Power (rook as mating piece) = 5.14-1 = 0.357
Sacrifice Material = ( ) ( )114 . 27 26 28 28− − − − = 0.071
Switchback = 1.2-1 = 0.5
Economy = ( ) ( )15 . 4.166 0.444 0− − + = 0.744
Total Score = Aesthetic Principles (3.003) + Themes (3.671) = 6.674
177
XABCDEFGHY
8-+-+-+-+(
7+-sN-+p+-'
6-+PzP-zP-+&
5+K+R+-+-%
4pzPp+kzp-+$
3trPzP-+r+-#
2LtRp+-vLQzp"
1+l+-+-+-!
xabcdefghy
XABCDEFGHY
8r+l+-mkntr(
7zp-wq-+-+p'
6-zpnzp-vlP+&
5+-+QzppvL-%
4-+-+-+-+$
3+NzP-zP-+-#
2PzP-+-zPP+"
1tR-+-mKLsNR!
xabcdefghy (a) Jukka Tuovinen, Thema Danicum,
1993 (score: 0.613) Soural, J. vs. Ruijgrok, D.,
Pardubice Czech op, 2007 (score: 0.444) 1. Bh4 Ke3 2. Rd4 Rf2 3. Qe4# 1. Bxf6 Be6 2. Qxe6 Qf7 3. Qxf7#
Figure 6.5 Lowest Scoring Combinations (a) COMP, (b) TG
The combination in Figure 6.5(b) scored 0.444 in total. It had four aesthetic principles
(the minimum, see subsection 6.1.2) and none of the selected themes. Economically, the
checkmate was quite poor and scored negatively. Its calculations are as follows.
Initial Position (POE = 0):
Sparsity = ( ) 1129 .86 1−
− + = 0.252
After 14. Bxf6 (POE = 1):
All of the Piece’s Power (bishop) = 1.13-1 = 0.077
After 15. Qxe6 (POE = 2):
All of the Piece’s Power (queen) = 1.27-1 = 0.037
178
After 16. Qxf7# (POE = 3):
Weakest Piece Possible to Checkmate (queen) = 4.27-1 = 0.148
All of the Piece’s Power (queen) = 1.27-1 = 0.037
Final Position (POE = 4):
All of the Piece’s Power (queen as mating piece) = 1.27-1 = 0.037
Economy = ( ) ( )115 . 1.166 0 3.333− − + = -0.144
Total Score = Aesthetic Principles (0.444) + Themes (0) = 0.444
Since it is possible that the evaluation of aesthetic principles (in which compositions
scored higher for all seven, see Experiment 2, section 6.2) might have been a significant
contributing factor here, this experiment was repeated but with aesthetic principles and
themes evaluated separately.
6.4.1 Aesthetic Principles Only
The cumulative evaluation of only the seven aesthetic principles was intended to see
how cumulative aesthetic assessment would fare missing the ten themes. The results are
as follows.
Table 6.8 Average Cumulative Aesthetic Scores for Aesthetic Principles Only
TG COMP Total
Set 1 Set 2 Set 1 Set 2 TG COMP 1.414
SD 0.488 1.429
SD 0.496 1.993
SD 0.564 1.997
SD 0.571 1.421
SD 0.492 1.995
SD 0.568
179
Compositions scored on average 0.574 points higher than tournament games for
cumulative aesthetic principle assessment, not including themes. This difference was
statistically significant (TTUV, 2T, SL 1%); t(29376) = 95.712, P<0.01. Between sets
within domains, the differences were very minor and not statistically significant.
Between the tournament game sets the difference was 0.015, t(12547) = -1.712, P =
0.087 and between composition sets it was 0.004, t(19339) = -0.475, P = 0.635. Figure
6.6 depicts the relationship between combinations and tournament games in total from
the perspective of the aesthetic principles only. The have been arranged in ascending
order for illustrative purposes.
Figure 6.6 Cumulative Scores Based on Aesthetic Principles Only
The highest score for aesthetic principles alone in a composition was 4.752 and for a
tournament game it was 3.950. The lowest were 0.613 and 0.444, respectively. It was
not unexpected that compositions would score higher in terms of aesthetic principles
0
1
2
3
4
5
1 1501 3001 4501 6001 7501 9001 105011200113501150011650118001
Scor
e
Positions
Compositions vs. Tournament Games
COMPTG
180
alone because they are more easily incorporated into compositions (see Experiment 2,
section 6.2).
6.4.2 Themes Only
Looking now at just the cumulative assessment of the ten themes, however (see Table
6.9), tournaments games scored higher than compositions by a small margin of 0.054
that was nonetheless statistically significant (TTUV, 2T, SL 1%); t(9666) = -8.35,
P<0.01. This is discussed in the following section.
Table 6.9 Average Cumulative Aesthetic Scores for Themes Only
TG COMP Total
Set 1 Set 2 Set 1 Set 2 TG COMP 0.682
SD 0.389 0.675
SD 0.404 0.614
SD 0.468 0.637
SD 0.471 0.679
SD 0.397 0.625
SD 0.470
The difference of 0.007 between sets in the domain of real games was not statistically
significant; t(4393) = 0.559, P = 0.576. Neither was the difference of 0.023 between the
composition sets; t(10126) = -2.446, P = 0.014. Figure 6.7 shows the distribution
graphically (arranged in ascending order). As mentioned in Experiment 1 (refer
subsection 6.1.1), themes only occurred in 10,132 compositions and 4,415 tournament
games.
181
Figure 6.7 Cumulative Scores Based on Themes Only
Even though the difference in means between composition sets was not significant at
the 1% level, if a significant difference was detected, it would not necessarily be a cause
for concern because human creativity is more prominent here than in tournament games
(and could therefore perhaps account for the detected difference between the means of
these particular composition sets; or the difference could simply be attributed to random
factors). A significant difference between real game sets, however, would have been
rather unexpected.
The incremental nature of the cumulative theme scores (Figure 6.7) is not as smooth
compared to the aesthetic principles shown in Figure 6.6. This is to be expected, given
the typically more dynamic nature of themes. Unlike the aesthetic principles, a mating
combination could have as little as one instance of a theme or up to twenty instances of
a collection of themes. The highest score in a composition for themes alone was 3.671,
-1
0
1
2
3
4
5
1 1501 3001 4501 6001 7501 9001
Scor
e
Positions
Compositions vs. Tournament Games
COMP
TG
182
and for a tournament game it was 4.000, whereas the lowest scores attained were -0.341
and -0.114, respectively.
6.4.3 Discussion
The results in Tables 6.8 and 6.9 suggest that cumulative assessment of aesthetic
principles alone is perhaps sufficient to capture the expected difference between the
domains of compositions and real games. Cumulative assessment of themes alone
demonstrates a difference between domains that is less prominent. While the aesthetics
of themes (common to both domains) is not something that is necessarily greater in
either one, in this case it was shown to be marginally higher for real games. It is quite
possible that in looking for a ‘common ground’ of themes, the selection was perhaps
inevitably slightly in favour of those found in real games (themes specific to or usually
associated with compositions are generally rare in real games and difficult to justify for
comparative purposes between domains), which might explain the result.
Even so, this possibility has not been confirmed and goes beyond the scope of this
thesis (average theme frequencies, in fact, might suggest otherwise, see section 6.1.1).
None of this, however, implies that the selection of aesthetic principles was
intentionally biased. That would only be obvious if they consisted specifically of
composition conventions or brilliancy characteristics (see section 3.3). Instead, they
were selected from both based on conformity to chess literature on the subject of
aesthetics in the game (see subsection 3.3.3 and Experiment 1, subsection 6.1.2).
Overall, this means that: 1) on average, compositions tend to contain more of these
aesthetic principles (in manifestations than translate to higher scores) and are thus more
183
beautiful in that respect, as might be expected; 2) cumulative theme assessment also
captures a difference between domains, even though not in favour of compositions (and
this would not necessarily be expected). It is likely more difficult to find these ten
themes incorporated into compositions than the (simpler to implement) aesthetic
principles.
These two factors therefore provide a certain balance to the overall aesthetic evaluation
of a combination. Such an evaluation is likely better represented using both aesthetic
principles and themes. Cumulatively they still translate to higher scores for
compositions, as expected (see section 6.4). The difference of 0.663 points between
domains is not only significant statistically, but also a prominent computational
difference. Aesthetic principle or theme assessment, alone, is unlikely to be as reliable.
6.5 Experiment 5: Conformity to Authoritative Human Assessment
Mastery of the game is not a prerequisite to aesthetic appreciation (see section 2.1).
However, a selection of beautiful games or compositions made by experts or renowned
authors in the area (for appreciation by also average players) is a reliable source for
comparative purposes. Obtaining personalized authoritative opinion of this nature was
beyond the financial means of the author and was, in any case, deemed less suitable
than what is available in chess literature. Actual publications on the subject are likely to
be better investigated and more reliable than simply asking experts (not all of whom
may have an eye for beauty in the game) which combinations they thought were among
the most beautiful.
184
This experiment was therefore performed to test if beautiful combinations in tournament
games (according to authoritative sources) would score higher aesthetically based on the
proposed model than combinations from regular tournament games. For this purpose,
three books featuring beautiful tournament game combinations were used: ‘Aesthetics
in Chess’ (Linder, 1981), ‘Anthology of Chess Beauty’ (Belov et al., 1996) and ‘Les
Prix de Beauté aux Échecs’ (Lionnais, 2002). The first is a collection of beautiful
compositions and tournament games by Isaac Linder, a renowned historian of chess.
This book was accepted in Italy as a ‘hymn to the beauty of combinations’ and copies of
it given to the participants of the First Congress of the Association of Italian Chess
Masters, held in Rome on 29 May 1982 (L'italia Scacchistica, 1982). Only the
tournament games were used.
The second is endorsed by Garry Kasparov himself (former world champion and
arguably the strongest player of all time) who wrote the foreword to that book. The third
is a well-known collection of beautiful tournament games by the mathematician and
specialist in the aesthetics of chess, Francois Le Lionnais (Osborne, 1964; Roubaud,
1998). This book has been in publication for nearly 70 years (the first edition was in
1939) and is also referenced in the second book. Other books on ‘amazing
combinations’ or ‘greatest games’ were not used because they do not explicitly use the
term ‘beauty’ to describe their content (and were therefore considered less reliable).
Most of the games from the selected books did not end in checkmate (as most
tournament games are resigned) but collectively, there were 61 games which could be
used (there were actually 67 games but six turned out to be repeated). In most cases,
each full game had to be entered manually into the computer (by playing out all the
moves) and converted into the PGN database format for analysis using CHESTHETICA
185
(see Appendix D). The task took a few days to complete because identification of usable
games from the books was time-consuming.
Extrapolation of forced mates (e.g. using a game engine) from games that did not
actually end in mate was not done because they would technically not be part of the
game considered beautiful by the authors. A forced mate could also be one of several
such variations possible at the point of resignation. Speculative selection in this case
would be even further away from what actually happened in the game. The 61
‘beautiful’ combinations were compared against combinations from 203 random
tournament games.
The tournament games selected were exclusively between grandmasters (Elo ≥ 2500) to
minimize bias. Games between or involving amateurs would likely be inherently less
beautiful due to unsound play. Note that all the available games between players rated
2500 Elo points and above, that ended in checkmate (from the tournament game data
set, see section 6.0), were used to get a better estimate of the true average aesthetic
value of those mating combinations. The 61 games were checked against these 203 to
ensure both sets were unique.
The 61 beautiful combinations scored a mean aesthetic score of 2.134 (SD 0.751)
whereas the grandmaster games scored a mean of 1.764 (SD 0.691). The difference of
0.37 was statistically significant (TTUV, 2T, SL 5%); t(93) = 3.442, P<0.05. Since no
‘levels’ of beauty were indicated for the beautiful combinations, positive correlation
with human assessment could not be demonstrated. However, this experiment suggests
that the proposed aesthetics model generally conforms to, or agrees with the judgement
of authoritative human sources concerning beautiful combinations in the game. This is
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especially significant considering that both collections were from the same domain (i.e.
real games), and given the rigorous comparison against grandmaster games as opposed
to those by amateurs. The following experiment tests for actual positive correlation with
human player aesthetic assessment.
6.6 Experiment 6: Correlation with Human Assessment
The previous 5 experiments demonstrate that the proposed model of aesthetics is able to
recognize aesthetics in the game meaningfully within the chosen scope of analysis. The
second part of experimentation was therefore to test the model for positive correlation
with human chess player assessment. This essentially seeks to determine if human chess
players tend to agree with the computational evaluations of beauty in the game.
With the cooperation of the online chess community, ‘Chessgames.com’ (Chess Games
Database, 2008) and funding from the author’s research grant (University Tenaga
Nasional research grant J510050123), four online (interactive) surveys involving
combinations from the data sets (see section 6.0) were conducted. The surveys were
open to members and visitors to the site. Given that each survey was relatively time-
consuming (taking about 20-30 minutes to complete), non-members were offered three
months premium membership to the site for their effort. Existing premium members
could take the surveys out of interest (and many did). An online chess community is
perhaps the best avenue for such surveys because players come from all over the world
and can participate at their own pace. The number of respondents is also theoretically
unlimited. The actual survey questions and results are available in Appendix F.
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Since it is difficult to demonstrate that a combination which scores say, 4.0 is exactly
twice as beautiful as one that scores 2.0, the computational scores were treated as
interval data. These can be used to differentiate between the levels of beauty in a
collection or database of combinations. Unlike ratio data with meaningful intervals and
a true starting point (i.e. zero), interval data have meaningful intervals but without a true
starting point (Donnelly, 2004). The capability of humans to differentiate between
combinations explains the wide availability of books featuring collections of ‘amazing’
combinations and ‘great’ games (Znosko-Borovsky, 1959; Fine, 1976; Burgess et al.,
2004; Hochberg, 2005; Sukhin, 2007; Levitt and Friedgood, 2008).
Each survey consisted of 20 randomly selected combinations. Respondents were asked
to rate them between 1 and 10 based on beauty and to ensure the ratings were precise to
one decimal point, which does not necessarily mean accurate to 0.1. It can also mean
accurate to just 0.5 or a whole number such as 7.0 (if they really felt that score best
reflected their assessment). This requirement was not explicitly stated for survey 2 (see
Survey 2, section 6.6.2). Preliminary testing showed that players tended to enter whole
numbers and as a result, many of the combinations were tied. The suggested level of
precision encourages respondents to decide one way or the other in such cases or enter
equal values only if they really feel that way. Respondents were free to modify their
ratings at any time before submission, e.g. after looking at all the combinations in the
survey. Preliminary testing also suggested that players found it difficult to look beyond
the fact that some of the combinations were not forced. It was difficult, they said, to
judge a combination as beautiful, otherwise.
While this is not a prerequisite for beauty in chess (see subsection 3.3.3), in the interest
of viable data, only forced combinations were used in all the surveys. The author
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wanted to avoid combinations being penalized for not being forced, which is what
otherwise might have happened and possibly tainted the experimental results. In any
real world application of aesthetic evaluation, available combinations can be filtered for
those with forced combinations, if necessary. There is no aesthetic evaluation of the fact
that a combination is forced, i.e. there are no meaningful degrees of it.
A combination is either a forced line of play or it is not. Incidentally, and to digress for
a moment, when comparing the average cumulative aesthetic score for the 2,233 forced
tournament game combinations and the remaining 10,319 that were not forced (from the
original data set, see section 6.0), it was found that the forced combinations scored
slightly higher with 1.767 (SD 0.670) compared to 1.637 (SD 0.601) for those that were
not forced. The difference of 0.13 was statistically significant (TTUV, 2T, SL 1%);
t(3059) = 8.474, P<0.01. It is possible that while a combination having a forced line of
play per se, is not a prerequisite to aesthetic appreciation, that ‘preference’ could have
some basis after all.
Returning to the surveys, only valid responses were used. Validation was determined
based on three criteria. First, a respondent had to answer two ‘control’ questions
correctly. It is difficult to confirm the official chess ratings of respondents, if any, so an
‘aesthetics competency test’ consisting of two questions was included in each survey
but not indicated as such (see Appendix F, section 1.4 for a better explanation). A
certain level of competence - but not necessarily mastery - is necessary to appreciate
beauty in the game (see section 2.1). There was a field for their most relevant chess
rating in the survey nonetheless. A good number of respondents answered ‘unrated’ to
this. Otherwise they ranged between 880 and 2496 (approximate Elo points, quite
possibly official), which covers the spectrum of chess playing strength quite well.
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Second, their evaluations should not appear suspicious such as having too many similar
values and no precision at all. This would suggest a hurried evaluation or even sabotage.
This is not as unlikely as it may sound. For instance, one chess grandmaster that the
author tried to consult with while designing the surveys refused to cooperate or offer
constructive feedback because he thought computers were becoming too intelligent for
the good of humanity. A conversation that ensued suggested he really believed this,
which the author found strange, especially for a grandmaster. Fortunately, the author
managed to find other chess experts who cooperated and offered valuable feedback to
improve the surveys (see Appendix F). Suspicious evaluations also included those with
even one combination rating outside of the specified range.
Third, repeat evaluations by the same person (in the same survey) were discarded and
only the most recent was used. The average human rating for each combination was
then calculated and compared against computational evaluation (also rounded to one
decimal point for consistency).
For the statistics, rp and rs represent the Pearson and Spearman correlation coefficients,
respectively. Interpretation of Spearman correlation strength can be subjective, so the
following categories were set based on the result first rounded to one decimal place.
0.0 < |rs| < 0.2 (very poor)
0.2 ≤ |rs| < 0.4 (poor)
0.4 ≤ |rs| < 0.6 (moderate)
0.6 ≤ |rs| < 0.8 (good)
0.8 ≤ |rs| ≤ 1.0 (very good)
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A ‘good’ correlation is also considered ‘strong’. Each of the surveys and analysis
performed on their respondent data is explained next. A comprehensive discussion of
the results is presented in subsection 6.6.5.
6.6.1 Survey 1 (Mixed)
The purpose of this survey was to see if human aesthetic assessment correlated
positively with computational assessment given a random selection of 20 direct mate-in-
3 composition and tournament game combinations (10 each and in no particular order).
There were a total of 134 respondents to the survey. Of these, 10 failed to answer the
competency control questions correctly. Another 6 were flagged as suspicious
evaluations and 2 had repeat evaluations. This left a total of 116 valid responses to be
analyzed. Table 6.10 shows the combination number (#), type, (average) human player
aesthetic assessment, standard deviation (SD) and computational aesthetic assessment.
Table 6.10 Human vs. Computer Assessment (COMP+TG Combinations)
# Type Human SD Computer # Type Human SD Computer 1 TG 3.9 1.9 1.1 11 TG 3.1 2.0 1.1 2 COMP 4.9 2.2 2.0 12 COMP 6.0 2.0 1.9 3 COMP 5.7 2.2 2.1 13 TG 5.5 2.1 1.0 4 COMP 6.4 2.2 2.9 14 COMP 6.1 2.2 2.0 5 COMP 4.8 2.2 2.3 15 COMP 4.9 2.5 1.6 6 TG 5.2 2.0 2.1 16 TG 5.8 2.0 1.0 7 COMP 5.4 2.3 1.5 17 COMP 4.8 2.0 2.2 8 TG 4.2 2.2 0.6 18 TG 5.6 2.1 1.5 9 TG 5.3 2.0 2.6 19 TG 4.5 2.2 1.5 10 COMP 5.9 2.2 2.3 20 TG 4.7 2.3 1.0
Correlation was tested using Spearman’s rank correlation (1T, SL 5%). Given the null
hypothesis (i.e. ρ≤0) and alternate hypothesis (i.e. ρ>0), a one-tailed test was deemed
the most suitable and sufficient for this experiment; ρ is the population correlation
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coefficient (not the calculated correlation coefficient, r). Spearman’s rank correlation is
a non-parametric measure of correlation which makes no assumptions about the
frequency distribution of the variables. It should be used when the distribution of the
data could make Pearson's correlation coefficient appear misleading. In effect, it is
usually a stronger test of correlation.
The Spearman rank correlation coefficient obtained was significant; rs = 0.408, t(18) =
1.9, P<0.05. Even though this correlation is not very strong, it nonetheless demonstrates
a moderate and statistically significant positive correlation between computational
aesthetic assessment based on the proposed model and human chess player aesthetic
assessment. This is especially significant given that the combinations used were selected
at random with sometimes very small differences in computational aesthetic evaluation
between them (e.g. 0.1, 0.2 points). The average computational score was 1.7 (SD 0.61).
In fact, the author was sceptical any significant positive correlation would be found at
all because such small aesthetic differences would be difficult for humans to
differentiate. Some respondents even left feedback suggesting as much. Incidentally, the
combinations they cited were mostly those taken from tournament games.
It is interesting that the human ratings for the 10 tournament game combinations alone
did not exhibit a significant positive correlation with the computational ratings (1T, SL
5%); rs = 0.093, t(8) = 0.26, P = 0.401. The human ratings for the 10 compositions alone
did not exhibit such as a correlation either (1T, SL 5%); rs = 0.104, t(8) = 0.3, P = 0.386.
This could be due to the smaller sample size (n=10) and restricted range of scores
(which usually affects correlation), compounded by the fact that the computational
ratings for some of the randomly selected combinations were not discrete enough (i.e.
distinct enough from each other).
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6.6.2 Survey 2 (Mixed, Discrete Evaluations)
The second survey tested combinations that were more evenly spaced across the
spectrum of computational assessment. This meant comparing combinations that were
factors of approximately 0.75 points apart. The range of aesthetic scores for these
combinations was therefore between 0.5 and 6.7 which is essentially between the lowest
and highest from the data sets (see section 6.4). This is unlike survey 1 (see subsection
6.6.1) where the computational scores ranged only between 0.6 and 2.9 with differences
as little as 0.1 points between them (or none at all). This second survey consisted of 10
pairs of combinations from compositions and tournament games (i.e. 5 pairs each, 20
combinations in total). Respondents were asked to choose the more beautiful
combination in each pair and then ascribe a rating between 1 and 10 to both. More pairs
would have been desirable but the survey had to be kept reasonable for the sake of data
viability and the respondents (some of whom complained that even 20 combinations
required “too much effort”).
Pairings were only between combinations from the same domain because chess players
can usually tell the difference and this might influence their assessment as to which in
each pair was more beautiful. Since paired combinations were used, it was unnecessary
to require precision to one decimal point. However, it was not explicitly forbidden and
some respondents entered precise values to one decimal point anyway. The first 5 pairs
were from tournament games and the following 5 pairs from compositions. In each, the
first pair was separated by 3.75 points, the second 3 points, the third 2.25 points, the
fourth 1.5 points and the fifth 0.75 points. The combinations were selected randomly as
long as these criteria were met. They were also arranged randomly so as to not have the
higher of the two on any one side of the page (indicating a pattern).
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The purpose of this survey was to see if a positive correlation existed between
computational evaluation and human assessment given a random selection of direct
mate-in-3 composition and tournament game combinations with discrete values i.e.
distinct differences between their computational evaluations. There was a total of 133
respondents to the survey. Of these, 25 failed to answer the competency control
questions correctly. Another 3 were flagged as suspicious evaluations. It is not known
exactly why 18.8% of respondents to this survey (compared to 7.5% in survey 1) failed
to answer the control questions correctly. It is possible that a larger number of weaker
players participated here. On the other hand, there were only half as many suspicious
evaluations for this survey. This left a total of 105 valid responses to be analyzed. Table
6.11 shows the combination number (#), type, (average) human player aesthetic
assessment, standard deviation (SD) and computational aesthetic assessment.
Table 6.11 Human Assessment vs. Computer (Discrete Evaluations)
# Type Human SD Computer # Type Human SD Computer 1 TG 6.8 2.1 4.5 11 COMP 4.8 2.1 2.9 2 TG 3.5 1.8 0.8 12 COMP 6.5 2.3 6.7 3 TG 7.1 1.7 4.5 13 COMP 6.0 2.2 3.4 4 TG 5.3 1.9 1.5 14 COMP 8.1 1.8 6.4 5 TG 3.2 1.7 0.5 15 COMP 6.8 2.1 3.7 6 TG 5.5 1.9 2.7 16 COMP 7.1 1.7 6.0 7 TG 3.8 1.7 0.5 17 COMP 6.4 2.3 4.4 8 TG 4.0 1.7 2.0 18 COMP 6.8 2.2 5.9 9 TG 6.1 1.8 3.5 19 COMP 7.7 2.0 5.0 10 TG 6.2 2.5 2.7 20 COMP 7.2 2.2 5.7
The Spearman rank correlation coefficient obtained was significant (1T, SL 5%); rs =
0.889, t(18) = 8.23, P<0.01. As expected, with discrete computational evaluations, a
stronger positive correlation with human assessment would be evident. The average
computational evaluation was 3.7 (SD 2). Using the same test, tournament game
combinations alone exhibited a very good and significant positive correlation with
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computational assessment; rs = 0.942, t(8) = 7.93, P<0.01. Compositions alone exhibited
a good positive correlation; rs = 0.59, t(8) = 2.07, P<0.05.
6.6.2(a) Levels of Agreement
The author also wanted to know if there was any correlation between ‘human-computer
agreement’ and the discrepancy between combinations in a pair. Essentially, this asks if
humans tended to agree more with computational assessment of beauty (as in which of
the two combinations in a pair was more beautiful) as the gap between both
combinations in pairs got bigger (in terms of computational evaluation). Levels of
agreement for both tournament game combination pairs and composition pairs were
analyzed together (n=10) since significance testing is more reliable with a sample size
of n≥6. Table 6.12 shows the combination number (#), the percentage of human
respondents whom agreed with the computational evaluation as to which of the two
combinations was more beautiful, and the difference in computational scores between
the two combinations. Tournament game combinations (#1-#10, 5 pairs) are on the left
and compositions (#11-#20, 5 pairs) are on the right.
Table 6.12 Level of Human Agreement with Computer Assessment
# % Agreement Diff. # %
Agreement Diff.
1 90.5 3.75 11 76.2 3.75 2 12 3 77.1 3.00 13 87.6 3.00 4 14 5 87.6 2.25 15 58.1 2.25 6 16 7 58.1 1.50 17 61.9 1.50 8 18 9 51.4 0.75 19 47.6 0.75 10 20
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A very good and statistically significant positive correlation was found (Pearson
correlation, 2T, SL 5%); rp = 0.835, t(8) = 4.3, P<0.01. This supports the idea that
prominent differences in computational evaluation tend to lead to better positive
correlation with human assessment (the Pearson test was used here because it was not
the individual combination ratings themselves being analyzed). Small differences in
computational aesthetic evaluation are therefore likely more difficult for humans to
perceive.
The ‘right amount’ or ‘minimum scale’ of beauty in terms of computational evaluation,
however, is difficult to determine. While very small differences are probably
undesirable, large differences are harder to attain. Any real-world application of the
computational model proposed might perform more reliably using a minimum scale of
1.0 when assessing combinations, so the results have a higher probability of concurring
with human assessment. Anything less than that might have people evenly split in their
agreement.
6.6.3 Survey 3 (Tournament Games)
The previous two surveys looked at composition and tournament game combinations
together. This was done because the proposed model of aesthetics focused on a
‘common ground’ applicable to both domains, not exclusively either one (see section
3.2). However, the author wanted to know what effect combinations from exclusively
either domain would have on respondent ratings and their correlation with
computational assessment. While a positive correlation might be expected here as well,
the author was sceptical because tournament games are not known for their beauty,
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unlike compositions. This would make aesthetic assessment problematic, especially
when there are no compositions present to compare against.
This third survey featured 20 randomly selected combinations from tournament games
only. The purpose of this survey was to see if a positive correlation existed between
computational evaluation and human assessment given a random selection of direct
mate-in-3 tournament game combinations. There were a total of 109 respondents to the
survey. Of these, 3 failed to answer the competency control questions correctly. Another
3 were flagged as suspicious evaluations and 1 had a repeat evaluation. This left a total
of 102 valid responses to be analyzed. Table 6.13 shows the combination number (#),
(average) human player aesthetic assessment, standard deviation (SD) and
computational aesthetic assessment.
Table 6.13 Human vs. Computer Assessment (TG Only)
# Human SD Computer # Human SD Computer 1 5.4 2.0 1.7 11 6.2 1.9 0.7 2 5.8 2.2 3.4 12 4.9 2.2 1.0 3 4.5 1.8 2.3 13 1.9 1.5 1.3 4 2.8 1.7 1.3 14 5.2 2.1 2.1 5 6.0 1.8 3.2 15 6.1 2.2 2.7 6 5.7 1.9 2.3 16 3.5 2.0 1.5 7 6.2 1.8 2.1 17 6.4 2.1 3.1 8 4.4 2.2 1.1 18 6.4 2.2 1.1 9 5.7 2.0 1.2 19 2.9 1.9 1.5 10 6.7 1.9 1.8 20 4.5 2.3 2.3
There was no significant positive correlation between human and computer assessment
using Spearman’s rank correlation test (1T, SL 5%); rs = 0.252, t(18) = 1.1, P = 0.143.
As suspected, when regular tournament games (with typically little aesthetic content)
are evaluated in terms of beauty, a significant positive correlation is unlikely to be
found. The problem is also compounded by the small differences in computational
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evaluation (some as little as 0.1 points) between some combinations. The average
computational evaluation was 1.9 (SD 0.8). This explains the very good correlation
attained in survey 2 (see subsection 6.6.2) where discrete computational aesthetic
ratings between tournament game combinations would have clearly separated the
beautiful from the bland for respondents.
6.6.4 Survey 4 (Compositions)
The fourth survey featured 20 randomly selected combinations from compositions only.
It stands to reason that since compositions are generally known for their beauty, it
would be more recognizable to respondents in this survey than tournament game
combinations (see previous subsection). The purpose of this survey was therefore to see
if a positive correlation existed between computational evaluation and human
assessment given a random selection of direct mate-in-3 compositions. There were a
total of 107 respondents to the survey. Of these, 25 failed to answer the competency
control questions correctly.
This relatively high number is likely due to the first control question. Beginners might
mistake the direction toward which the pawn on g2 can capture (see Appendix F, Figure
F.7(a)). A handful of respondents actually commented as much (in the survey feedback
box), thinking the provided solution was flawed. One response was flagged as a
suspicious evaluation. This left a total of 81 valid responses to be analyzed. Table 6.14
shows the combination number (#), (average) human player aesthetic assessment,
standard deviation (SD) and computational aesthetic assessment.
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Table 6.14 Human vs. Computer Assessment (COMP Only)
# Human SD Computer # Human SD Computer 1 4.9 2.5 1.4 11 4.9 2.5 2.7 2 4.3 2.2 2.0 12 6.2 2.5 3.8 3 5.3 2.1 2.4 13 4.0 2.0 1.5 4 5.7 2.0 3.5 14 5.8 2.4 3.5 5 7.6 1.7 3.3 15 5.5 2.4 1.8 6 5.2 2.4 1.8 16 5.1 2.3 1.5 7 5.4 2.1 2.3 17 5.4 2.1 3.3 8 4.2 2.4 1.1 18 4.2 2.3 2.8 9 5.3 2.5 1.4 19 5.7 2.2 2.3 10 5.0 2.3 1.2 20 5.8 2.3 1.9
Correlation was tested using Spearman’s rank correlation (1T, SL 5%) and was found to
be significant; rs = 0.584, t(18) = 3.05, P<0.01. This is a good positive correlation that
supports the idea that where beauty is prominent, it is more easily recognized and rated
by humans (unlike in survey 3, see subsection 6.6.3). The problem of minute differences
between compositions (in terms of computational evaluation) would likely still have had
some effect on the results (contrast with survey 2, see subsection 6.6.2). The average
computational evaluation score in this survey was 2.3 (SD 0.8).
6.6.5 Survey Conclusions
The following table summarizes the results of Experiment 6; S = significant, NS = not
significant, LOA = level of (human) agreement, LOD = level of (computational)
distinction between scores, n = sample size, rs = Spearman rank correlation coefficient,
and rp = Pearson correlation coefficient. All Spearman tests were for positive correlation
with computational assessment (1T, SL 5%). The Pearson test was for any correlation
(2T, SL 5%).
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Table 6.15 Summary of Human-Computer Assessment Correlations
Survey 1:
Random Ratings TG+COMP (n=10+10)
Survey 2: Random (Discrete)
Ratings TG+COMP (n=10+10)
Survey 3: Random Ratings
TG (n=20)
Survey 4: Random Ratings COMP (n=20)
S
rs=0.408, t(18)=1.9, P<0.05
S
rs=0.889, t(18)=8.23, P<0.01
NS
rs=0.252, t(18)=1.1, P=0.143
S
rs=0.584, t(18)=3.05,
P<0.01
TG (n=10)
NS
rs=0.093, t(8)=0.26, P=0.401
COMP (n=10)
NS
rs=0.104, t(8)=0.3, P=0.386
TG (n=10)
S
rs=0.942, t(8)=7.93,
P<0.01
COMP (n=10)
S
rs=0.590, t(8)=2.07,
P<0.05 LOA vs. LOD
S
rp=0.835, t(8)=4.3, P<0.01
The following is evident from the results of the surveys with regard to positive
correlations between average human chess player assessment of aesthetics and
computational aesthetic assessment (based on the proposed model).
1. There is a moderate positive correlation between human and computer
assessment for randomly selected combinations from both domains (real games
and compositions) when evaluated together by humans despite the presence of
very small differences in computational evaluation between certain
combinations.
2. There is no significant positive correlation between human and computer
assessment when looking at either domain separately in the scenario as
described in (1).
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3. There is a good and significant positive correlation between human and
computer assessment for randomly selected combinations with discrete
computational evaluations from both domains. There is also a very good and
significant positive correlation between the level of human agreement and the
level of distinction between combinations.
4. There is a very good and significant positive correlation between human and
computer assessment when looking at just the domain of real games in the
scenario as described in (3). There is a good and significant positive correlation
between human and computer assessment when looking at just the domain of
compositions in the scenario as described in (3).
5. There is no significant positive correlation between human and computer
assessment for randomly selected combinations from the domain of real games
alone with the presence of very small differences in computational evaluation
between certain combinations.
6. There is a good and significant positive correlation between human and
computer assessment for randomly selected combinations from the domain of
compositions despite the presence of very small differences in computational
evaluation between certain combinations.
Table 6.16 presents an interpretation of the information in Table 6.15 and points (1)
through (6) above.
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Table 6.16 Summary of Positive Correlations
TG Indistinct
COMP Indistinct
TG Distinct
COMP Distinct
n=10 (subsample)
Very Poor
Very Poor
Very Good Good
n=20 (mixed) Moderate Very Good
n=20 (exclusive) Poor Good
With small subsamples (n=10), computational aesthetic evaluation in both domains fair
very poorly in terms of positive correlation with human aesthetic assessment where
there are combinations with indistinct computational differences between them.
However, taken together with a combined sample size (n=20), both domains correlate
moderately with human assessment despite those small differences. This suggests that
each domain complements the other by providing a kind of scale or contrast which
might enable humans to rate them more accurately (as a whole and given a sufficient
sample size).
A larger sample size (n=20) in an exclusive setting appears to improve the positive
correlation for both domains despite indistinct computational differences between
combinations. The improvement is still inadequate for tournament game combinations
but significant for compositions. This suggests that aesthetic content is also factor.
Where beauty is not particularly present (as in the domain of real games) it probably
becomes more difficult for humans to rate them in terms of beauty. To digress for a
moment, the average number of pieces in the initial positions of the combinations for
survey 3 (TG) was 16.5 compared to 14.35 for survey 4 (COMP). The difference was
not statistically significant (TTUV, 2T, SL 5%); t(37) = 1.283, P = 0.208. Therefore the
number of pieces on the board per se is probably not a reason which influences
respondent ability to rate combinations aesthetically (as the author initially thought).
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Given distinct computational differences between combinations (i.e. discrete computer
ratings), both domains correlate quite well either as small subsamples (n=10) or mixed
together (n=20). Tournament game combinations appear to benefit the most in terms of
improved correlation from having these distinct differences. From Table 6.16, it is fair
to assume that in an exclusive setting (n=20) and given distinct differences in
computational evaluation for the combinations, both domains would also correlate as
well with human assessment. Given the author’s limited resources, all possible
permutations of these surveys could not be tested and some inferences are therefore
necessary.
Finally, using all available data from all four surveys including both discrete and
indiscrete computational ratings for both domains (n=80), the computational aesthetic
evaluations correlated strongly and positively with human aesthetic ratings and was
statistically significant (Spearman rank correlation, 2T, SL 1%); rs = 0.648, t(78) = 7.52,
P<0.01. Based on the experimental results in section 6.6 and the explanations presented,
the author may conclude that computational assessment based on the proposed model of
aesthetics does indeed correlate positively with human aesthetic assessment. Results are
better between combinations that have discrete computational aesthetic evaluations (e.g.
a difference of at least 1 point between them).
6.7 Chapter Summary
The first experiment demonstrated that the selected themes and aesthetic principles were
not exclusive to either domain and consistent with the proposed conceptual framework
for aesthetics (see section 3.2). Based on the formalizations developed for the seven
aesthetic principles and ten themes (see section 3.4), the second and third experiments
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demonstrated that there was a statistically significant difference for each of them (with
the exception of two themes, see section 6.3 for a discussion) between domains but not
within domains. Given that the board, pieces, rules, move length and objective (i.e.
checkmate) were the same for combinations in both domains, it was reasonable to
conclude that this detected difference was most likely aesthetic in nature, and could be
captured computationally using the formalizations developed (following the proposed
aesthetics model, see section 3.1).
The fourth experiment was based on the cumulative assessment of all seven aesthetic
principles and ten themes. It showed that compositions, on average, scored higher
aesthetically than tournament games, to a statistically significant degree. This is
consistent with what we know about the domains and further validates the aesthetic
recognition capabilities of the model. It is also notable that compositions, on average,
scored higher for aesthetic principles alone; but with themes alone, tournaments games
scored marginally higher aesthetically (see subsection 6.4.3 for a brief discussion on
this).
Experiment 5 tested the model for conformity to authoritative human assessment of
beauty (based on three books on chess beauty). Combinations from games deemed
beautiful by the authors scored, on average, higher based on the model than those from
regular tournament games between grandmasters (as opposed to amateurs which would
have been a less rigorous test). The difference was also statistically significant. The
results of experiments 1 through 5 therefore suggest that the proposed model of
aesthetics is able to recognize beauty in the game (within at least the scope of mate-in-
3). So in answer to research question 1 (see section 1.6): Can aesthetics in chess (within
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a specific scope) be recognized computationally? The author may conclude that, yes it
can.
Experiment 6 was performed using data from four interactive online surveys with
human chess players, as respondents. In general, there was a positive correlation
between computational aesthetic assessment (based on the proposed model) and human
aesthetic assessment, for both domains. However, the results also suggest that
correlation tends to improve when the combinations have discrete computational
evaluations (e.g. a difference of 1 full point between them).
Correlation is also probably better using compositions, due to their more prominent
aesthetic content, which is easier for humans to discern and rate compared to
tournament game combinations. In general, combinations with aesthetic scores around
1.5 and below would tend to resemble those found in tournament games (i.e. less likely
to be considered beautiful) whereas those with scores around 2.5 and above would tend
to resemble compositions (i.e. more likely to be considered beautiful). These values are
approximations based on the results shown in section 6.4, Table 6.7. Cumulatively
(using data from all the four surveys), there was a statistically significant and strong
positive correlation between computational aesthetic evaluation and human aesthetic
evaluation. So in answer to research question 2 (see section 1.6): If so, do the
computational evaluations correlate positively with human chess player aesthetic
assessment? The author may conclude that, yes they do.
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CHAPTER 7: CONCLUSION
7.0 Preliminary
This chapter presents a summary of the work in this thesis, with emphasis on the
contributions, implications of the research and possible directions for further work. Part
of the work presented in this thesis has already been cited as being a new area of games
research (van den Herik, 2006, 2008). Therefore, there may be many implications and
directions for further work; not all of which the author would have thought of, or
mentioned here. While the author himself will likely pursue some of these directions in
future research projects, there is much that other researchers can also do to explore the
area further and extend our knowledge of it.
7.1 Thesis Summary
This research began by looking at the problem of evaluating aesthetics in the game of
chess. Beauty in the game is appreciated by players (and composers) yet computers
were still unable to tell the difference between a beautiful game or combination and a
bland one. There are two notable reasons for this. First, current practices of addressing
aesthetics in the game (e.g. in automatic problem composition) conflate aesthetics with
composition convention, thereby failing to account for the former adequately. Brilliancy
in real games for example, demonstrates that aesthetics or beauty is perceived not only
in compositions. Second, is the use of fixed values to represent chess themes or related
principles (to account for aesthetics). These do not cater for the variety of configurations
possible on the chessboard.
206
A conceptual framework was therefore proposed to facilitate proper investigation into
aesthetics. Accordingly, aesthetics is treated as a component not exclusive to either the
domain of compositions or real games, yet consistent with chess literature on the
subject. This provides a way of thinking about aesthetics as an independent component
that can be applied, as required, to either domain. Aesthetics in the game was examined
from various angles including composition conventions, brilliancy characteristics (in
real games) and the principles of aesthetics as described in the relevant literature. For
experimental purposes, a selection of seven (refined) aesthetic principles and ten
(common) chess themes was made. These were used to represent the ‘common ground’
of aesthetics described in the framework.
A formula for cumulative aesthetic assessment was presented. It was based on the idea
that the more aesthetic principles and themes there are in a combination, the more
beautiful it is likely to be. However, each aesthetic principle and theme was represented
using a specifically designed and dynamic evaluation function (also called a
formalization). Therefore, a higher concentration alone, of aesthetic principles and
themes (in a combination), would not necessarily guarantee a high cumulative aesthetics
score.
Each of these evaluation functions used as parameters metrics and properties inherent to
the game. Metrics include the pawn unit, piece unit and board squares. These form the
basis of higher properties such as piece value, piece count, distance, piece power,
mobility and piece field. The advantage of these metrics and properties is that they are
widely accepted amongst chess players. While they are themselves perhaps not directly
related to aesthetics per se, they can be used to estimate aesthetic content in a
207
combination. The key lies in using them to map or translate the relevant knowledge we
have on aesthetics in chess literature into computational form.
This is analogous to expert knowledge on game-playing which has been translated into
the form of computational heuristics, with promising results. However, in game-playing,
search capabilities are equally if not more important than knowledge. A general
methodology for developing aesthetic evaluation functions was also presented with the
notable concept of the (aesthetic principle, and theme) ‘benchmark’. This typically
provides the range necessary in each function to differentiate between one
implementation (i.e. instance) of a principle or theme, and another.
For experimental purposes, aesthetic evaluation was limited to the scope of mate-in-3
combinations. Most of the aesthetics formalizations themselves are applicable
individually and can go beyond the scope of mate-in-3, but it would have been difficult
to demonstrate aesthetic recognition capabilities and positive correlation with human
aesthetic assessment if a mixture of shorter and longer combinations was included in the
scope. It would have been even more difficult to do so with no specific achievement at
the end (e.g. checkmate). The chosen scope is nevertheless considered reasonable in the
game; and there is an abundance of compositions and real game combinations available
that conform to it.
Experimentation was done in two stages. The first was to answer the first research
question, i.e. Can aesthetics in chess (within a specific scope) be recognized
computationally? Five novel experiments comparing compositions and real games
demonstrated that it can, based on the proposed model of aesthetics. The next stage was
designed to answer the second research question, i.e. If so, do the computational
208
evaluations correlate positively with human chess player aesthetic assessment?
Experiments on the data gathered from four online surveys (with hundreds of
respondents) showed that, overall, the computational evaluations did indeed correlate
well and positively with human chess player aesthetic assessment. However, the results
are likely to improve if the computational evaluations for the combinations are discrete,
i.e. there are prominent distinctions between them. This is because it is probably
difficult for humans to differentiate between minor aesthetic differences.
7.2 Thesis Contributions
This thesis makes the following contributions.
1. A conceptual framework for aesthetics in chess.
2. A review and examination of aesthetic principles (and themes) in the game.
3. A formula for cumulative aesthetic assessment in a move combination.
4. A general methodology for developing aesthetics formalizations in the game.
5. Seven dynamic evaluation functions for seven aesthetic principles in the game.
6. Ten dynamic evaluation functions for ten themes in the game.
7. A set of novel experiments for testing aesthetic recognition in the game.
8. A computer program that facilitates aesthetic evaluation based on the proposed
model.
9. A computational model for recognizing aesthetics in the game in a way that
correlates positively with human aesthetic assessment.
Contributions 1 through 4 are supported by logical arguments based on the relevant
literature. Contributions 5 and 6 were designed specifically by the author for this
209
research. They are based on metrics and properties inherent to the game. Even though
contributions 1 through 6 are constituents of the proposed aesthetics model, they are all
discrete steps or components that can be used creatively and independently (with the
exception of perhaps two theme formalization, see section 6.3). Contribution 7 provides
a way of actually testing the capabilities of computational aesthetic models or
approaches in the game. Previously, only the endorsement of one or two experts if any,
were used as validation. Contribution 8 embodies the proposed model (including the 17
formalizations) and makes experimentation and aesthetic analysis feasible. It is also
available to other researchers.
Contribution 9 is the proposed model of aesthetics itself which, following all its discrete
components, enables computers to recognize aesthetics in the game (within at least the
scope of direct mate-in-3) in a way that correlates well with human chess player
aesthetic assessment. This was confirmed through 6 experiments, the last of which
included the use of 4 surveys. The actual combinations used in the surveys and
corresponding respondent aesthetic ratings themselves (see Appendix F, sections 1.2
and 1.5) could also prove useful as experimental data sets for other researchers in the
area. However, this is not explicitly listed as a thesis contribution.
7.3 Implications of the Research
This research has shown that computational evaluation of aesthetics in the game of
chess is not only possible but also generally reliable within a reasonable scope. This
might go against preconceived notions or old adages such as ‘beauty is in the eye of the
beholder’. In light of the reasonable progress that has been made with regard to
aesthetics in other domains such as art and music, it is not surprising that results are as
210
good (if not better) for a zero-sum perfect information game like chess, given its nature.
The ‘mechanics’ behind aesthetic appreciation in the game are no mystery. Human
players have essentially known what they are for some time and the evidence is in the
literature. While there may be many ‘dimensions’ of beauty to the human eye that
perhaps cannot be exactly quantified, there is no scientific evidence to suggest that the
sort of approximation of beauty achieved in this research cannot account for ‘true’
beauty partially or indirectly, and in a comparative way that is generally consistent with
human aesthetic perception.
Analogously, the aesthetic principles and themes evaluated in this research might not
necessarily exclude the beauty of more complex ones not explicitly accounted for.
Unfortunately, it is difficult to demonstrate if the dynamics of more complex aesthetic
principles and themes are somehow dependent upon simpler ones, such as those used in
this research, i.e. the former cannot or seldom occur otherwise. If so, this would imply
the need of only a ‘critical set’ for aesthetic evaluation.
On the other hand, formalizations for aesthetic principles and themes different from
those used in this research (if developed using the same proposed methodology) should
exhibit similar experimental results. Given that a ‘common ground’ of aesthetics
between compositions and real games has been examined and tested, it is difficult to
imagine otherwise unless there was an unintentional but significant bias toward either
domain by using a different set of principles and themes. This could very well happen if
one is not careful to differentiate (at least to a reasonable degree) between composition
convention, brilliancy characteristics and the area in which they overlap. An interesting
application of the proposed model, as it stands, might be for computers to effectively
simulate the human player ability to often ‘sense’ the difference between a real game
211
combination and an artificial or composed one. This would nevertheless require further
experimentation.
There is probably no imperative to ‘reinvent the wheel’, e.g. by looking for other
aesthetic principles aside from what has been repeatedly described in chess literature,
and weighting them based on certain data sets (if any). The fundamental metrics and
properties of the game are also sufficient as building blocks for representing aesthetic
principles and themes, formally. Meaningful complexity (such as beauty) can, perhaps,
emerge from the relationships between much simpler components that may themselves
have little or nothing to do with aesthetics (Gell-Mann, 2007). This implies that if
analogous metrics and properties could be found or established in other domains (even
beyond chess variants and other zero-sum perfect information games), the approach
used in this thesis might be applicable there as well.
Even so, personal taste in aesthetics is still relevant and can be accommodated
computationally. There are at least two ways. First, using the proposed model, more
weight (e.g. in the form of multipliers) can manually be associated with the
formalizations for certain aesthetic principles or themes to suit a particular user’s
aesthetic preferences. Second, an aesthetics computer program incorporating the model
could train itself over a period of time based on say, a log of the user’s selection of
preferred aesthetic combinations. Using this information, it could automatically present
combinations that would most likely appeal to said user. Looking even further, a
computer could perhaps be designed to possess or develop a taste of its own, however
rudimentary its fundamental design was. For instance, it has been shown that human-
like features exhibited by a machine, tend to lead humans to believe it has a ‘mind’
(Krach et al., 2008). This would likely improve human-computer interaction.
212
Enabling computers to recognize aesthetics in games and other domains should not be
seen as a threat to humanity, or necessarily as a step toward simulating ‘feelings’ in
machines. While the latter may be a potential application, the main intention is usually
for the immediate benefit of humans. Computers, in principle, can analyze and ‘see’
much more (at least in terms of volume) than humans ever could in their lifetimes. A
great deal of this information (be it chess combinations, photographs or music) is
disregarded by machines in their typical tasks. It would be of conceivable benefit to
humans, for example, if computers could identify items that might be of aesthetic
interest to them. In the game of chess, at least, aesthetics recognition technology can be
used as an additional tool for ‘data mining’ aesthetic and educational material from
growing tablebases (see section 2.4).
7.4 Directions for Further Work
Further research endeavours in the area might look into the following.
1. Precision of computational aesthetic assessment.
2. Themes with no aesthetic distinction.
3. Shorter, longer and inconclusive move sequences.
4. Enhanced automatic problem composition.
5. Aesthetics in chess variants and similar games.
6. Aesthetic evaluation functions as game heuristics.
The first issue has to do with precision of computational aesthetic assessment. This
research focused on aesthetic recognition in the game of chess within the scope of direct
mate-in-3 combinations. While a strong positive correlation with human aesthetic
213
assessment was demonstrated, the level of precision for computational aesthetic
evaluation that correlates best with human assessment was not, due to resource
limitations. Discrete aesthetic scores for combinations, e.g. with differences of at least 1
point between them was suggested but this remains to be confirmed experimentally. So
does the ‘minimum’ computational score that qualifies as ‘beautiful’ (according to
human players).
Precision might help establish a reliable scale (though more in the ordinal or interval
rather than ratio sense) for aesthetics in the game that can be used to assist judges in
composition tournaments and to award brilliancy prizes to real games. Even though the
maximum aesthetic score attained for a combination from the 31,896 analyzed was
6.674, it is an open question whether this is the aesthetic ‘ceiling’ for a combination or
even close to it. It would be interesting to demonstrate (perhaps even mathematically)
what the ceiling, based on the proposed model, might be; and even more so, the sort of
combination that would qualify. Beauty in chess has, after all, been described as having
its basis in pure mathematics (Newman, 2003).
As a rough estimate of the highest aesthetic score, the maximum a combination could
score based on the model would be around 17 points (ignoring multiple instances of
themes), given that each aesthetic principle and theme has a theoretical limit of 1 (see
subsection 3.5.3). However, that limit can sometimes be exceeded. Also, it is unlikely
that a single combination could contain all 17 aesthetic principles and themes. The
actual aesthetic ‘ceiling’ would therefore be constrained by what is possible in the
game; a value that is probably quite different from the rough estimate.
214
An analogous situation can be found in Scrabble®. The highest score attained in a
single turn in an actual game (using the Official Scrabble Players' Dictionary) is 365 for
the word ‘quixotry’ (Fatsis, 2006) whereas the highest so far in theory, is 1682 for the
word ‘demythologizers’ (Word Records, 2008). Another aspect related to precision (that
perhaps psychologists might wish to explore) is the relationship between chess expertise
and aesthetic perception in the game. For instance, do experts derive more aesthetic
pleasure from the game than novices and average players? This could relate back to the
proposed aesthetics model and help refine it in certain ways.
The second issue to pursue further is themes with no aesthetic distinction between
domains. At least two were found in the course of this research (see section 6.3).
Assuming that their formalizations were not inadequate (they were, after all, developed
using the same methodology as the other eight), it could be that certain chess themes
inherently do not exhibit aesthetic differences between domains like most of the others.
The question then, is why? Another set of novel experiments will likely be required to
answer this.
The third issue is shorter and longer move sequences in the game. Shorter move
sequences (e.g. 1 move) are more convenient to assess aesthetically and this is useful in
say, deciding when to generate automatic chess commentary for a game. However, if the
move does not have a clear achievement such as checkmate, it would be difficult for it
to qualify as worthy of aesthetic assessment. Nevertheless, human players do tend to
find some single moves like that particularly beautiful and such criteria would be worth
investigating (Krabbé, 1998). Many of the formalizations proposed in this thesis can, in
principle, already be used to gauge the aesthetics of single, non-checkmate moves.
215
Longer combinations such as mate-in-4 and mate-in-5 are harder to find and more
difficult to assess aesthetically because they are usually more complex. The same
amount of (say, computational) beauty in a shorter combination would likely be
appreciated more by human players because the combination is easier to understand and
assess. The approach of using instead, the average amount of beauty per move (i.e.
‘aesthetic score/moves’), while worth testing, is unlikely to remain consistent as the
combinations get even longer (e.g. 10 moves).
Long combinations with inconclusive or ambiguous achievements such as gaining
positional superiority or forcing a draw may not be worthy of aesthetic assessment
either. However, forcing a draw when one is losing or behind in material can be
construed as a tangible achievement that warrants aesthetic assessment. Aesthetic
prerequisites for inconclusive or less distinct achievements (especially in long
combinations) can therefore be determined and tested. It would be interesting to know
for example, if non-checkmate achievements are seen as equally or more beautiful than
checkmates.
The fourth issue is automatic chess problem composition. It could likely progress
further using the proposed model. Existing approaches could be enhanced by simply
incorporating aesthetic assessment in addition to their assessment of composition
conventions. It would be interesting to see if the quality of computer compositions will
then improve. Perhaps even to a level comparable to human composers at the
competitive level.
The fifth issue has to do with chess variants and similar games. Aesthetics is also
perceived in other variants of chess and similar (as in zero-sum perfect information)
216
games such as Go. The same basic model can be applied to them (see section 3.1). It
remains to be demonstrated, however, if the results are as encouraging. It is possible
that the aesthetics of even non-deterministic games (e.g. backgammon) can be estimated
based on the model, if there is sufficient literature on the subject (for reference). This is
because aesthetic evaluation need not necessarily factor in the element of chance in
such games. A direct adaptation of the model could perhaps be tested on a non-
deterministic variant of chess (Develin and Payne, 2008).
The sixth issue is in terms of game-playing heuristics. The 17 aesthetic evaluation
functions - 7 aesthetic principles and 10 themes - developed for this research (or
adaptations of them) could be used as viable game heuristics in place of, or in addition
to, standard ones (e.g. see section 2.6). There is some potential for example, in using
aesthetic heuristics to solve complex chess problems. Game engines using standard
heuristics are likely to miss these solutions because they do not examine every possible
move, and typically bypass unusual ones.
217
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227 Znosko-Borovsky, E. (1959). The Art of Chess Combination, Dover Publications
Inc., New York, N.Y.
233
APPENDIX A: CHESS RULES
The following sections describe the relevant information pertaining to the rules and
notation used in the game of international chess. Rules and figures are from ‘The Chess
Variant Pages’ (Bodlaender, 2000). The rules of the game were edited for clarity and to
incorporate corrections based on FIDE (2005); a comprehensive explanation of the laws
of chess is available in the same source.
1.0 Introduction to the Game
Chess is a game played by two players using light and dark pieces (typically white and
black). The game is played on a chessboard, consisting of 64 squares with eight rows
and eight columns. The squares are alternately light and dark as well. The board must be
laid down such that there is a light square in the lower-right corner. The pieces used are
as shown in Table A.1.
Table A.1 The Chessmen
Chessman Name Symbol K king K Q queen Q R rook R L bishop B N knight N P pawn P
The symbols might differ in certain publications depending on the set type. Each player
must move one piece at a time, with White to start. A player may capture an opponent’s
piece by moving one of his own pieces to that square. The opponent’s piece is then
234
removed from the board. The objective of each player is to checkmate his opponent.
That means putting the enemy king in such a position that when attacked it cannot avoid
capture on the next move. When a player checkmates his opponent’s king he wins the
game. Both players develop their positions by making various moves and capturing
various pieces. This means that the number of pieces will diminish as the game
progresses. There are times when neither side can win, and the game is drawn.
Each player has sixteen pieces to start: one king, one queen, two rooks, two bishops,
two knights, and eight pawns. Figure A.1 shows the chessboard with all the pieces in
their initial position.
Figure A.1 The Initial Position of the Pieces
1.1 Movement of the Pieces
In this section, the movement of each piece type is explained. The more complex piece
movements are explained later.
235
1.1.1 Rook
The rook moves any number of squares in a straight line, horizontally or vertically
(Figure A.2) and may not jump over other pieces.
Figure A.2 Movement of the Rook
1.1.2 Bishop
The bishop moves in a straight diagonal line (Figure A.3) and may not jump over other
pieces.
Figure A.3 Movement of the Bishop
236
1.1.3 Queen
The queen has the combined moves of the rook and the bishop and may move in any
direction (Figure A.4).
Figure A.4 Movement of the Queen
1.1.4 Knight
The knight makes a move that is best described as an ‘L-shape’ (Figure A.5). Therefore,
it can ‘jump’ over other pieces unlike the bishop, rook and queen.
Figure A.5 Movement of the Knight
237
1.1.5 King
The king moves one square in any direction (Figure A.6). There is also one special type
of move made by a king and rook simultaneously called ‘castling’ (see the following
subsection). The king is the most important piece in the game and moves must be made
in such a way that the king is never in check (i.e. put on a square that is controlled by an
enemy piece).
Figure A.6 Movement of the King
1.1.5(a) Castling
Castling is usually a defensive manoeuvre intended to secure the king quickly against an
enemy attack. It involves moving the king and either rook simultaneously. The
following conditions must be met for castling to be permitted.
• The king has not yet moved in the game.
• The rook has not yet moved in the game.
• The king is not in check.
238
• The king does not cross a square that is attacked by an enemy piece.
• The king does not move to a square that is attacked by an enemy piece.
• All squares between the rook and king are empty.
In castling, the king moves two squares towards the rook and the rook moves over the
king to the next square (Figure A.7). In the diagram, Black's king on e8 and rook on a8
move to c8 and d8, respectively (i.e. long castling or queenside castling, Figure 7(a)).
(a) (b)
Figure A.7 Before and after Castling
Figure A.8 Castling Illegal for Both White and Black
239
White's king on e1 and rook on h1, move to g1 and f1, respectively (i.e. short castling or
kingside castling, Figure 7(b)). Figure A.8 shows a situation where castling is illegal for
both sides.
1.1.6 Pawn
The pawn is the weakest piece on the board and moves differently from the other pieces.
From its starting square, it has the choice of moving one or two squares forward as long
as the path is clear or one square to the left or right when capturing an enemy piece
(Figure A.9). If it is not on its starting square, then it can only move one square forward
if the path is clear or capture an enemy piece to the right or left.
Figure A.9 Movement of the Pawn
There is also a special rule called ‘en passant’ (French for ‘in passing’). When a white
pawn for example, makes a double step from the second row to the fourth row and there
is an enemy pawn on an adjacent square on the fourth row, then this enemy pawn may
in the next move capture diagonally to the square on the third row that was passed over
by the double-stepping pawn. In this same move, the double-stepping pawn is taken
240
(Figure A.10). The ‘en passant’ manoeuvre must be done immediately or it becomes
illegal. The same applies to Black.
Figure A.10 En passant
Pawns that reach the last row of the board must be promoted to (i.e. replaced by) any
piece except the king. Usually players will promote the pawn to a queen, but the other
types of pieces are also allowed. Promotions can result in pieces exceeding the standard
set (e.g. three rooks or two queens).
1.2 Check, Checkmate and Stalemate
A player is said to be in check when his king can be captured on the next move. For
instance, White moves his rook to a position such that it attacks the black king, i.e. if
Black does not do anything about it the rook could take the black king in the next move
(Figure A.11). It is considered good manners to say ‘check’ when one performs the
manoeuvre. It is also illegal to make a move such that one’s king is in check after the
move.
241
Figure A.11 Check
When a player is in check and he cannot make a move such that after the move, the king
is not in check, then he is checkmated or mated for short (Figure A.12(a)). The player
who is mated has lost the game. There are three ways to remove oneself from a check.
• Move the king away to a square where he is not in check.
• Capture the piece that gives the check.
• Move a piece between the checking piece (if it is a queen, rook or
bishop) and the king.
When a player has no legal moves but is not in check, then the player is said to be
stalemated (Figure A.12(b)). In a case of a stalemate, the game is drawn.
242
(a) (b)
Figure A.12 Checkmate and Stalemate
1.3 Other Rules
The following are explanations of other rules that are important in the game.
1.3.1 Resignation and Draws
A player can resign the game at any time, meaning that he loses and his opponent wins.
This is traditionally done by tipping the king over with one’s finger. A player may also
propose a draw after making a move and it is up to his opponent to accept or decline
before continuing the game. There is currently an ongoing debate in the chess
community about the ruling on mutually agreed draws (Milener, 2008).
It started primarily because chess audiences were disappointed with the increased
frequency of such draws (especially short and seemingly peaceful ones) between master
players in tournaments. These often resulted in ‘uninteresting’ games which are
typically unbecoming of master players. One of the most interesting suggestions was for
a draw offer to be valid throughout the rest of the game, discouraging premature
243
proposals (Karen, 2007). The debate may result in the FIDE changing the ruling on
draws in the future.
1.3.2 Repetition of Positions
If the same position with the same player to move is repeated three times in the game
(not necessarily consecutively), the player to move can claim a draw. However, when
the right to make a certain castling move is lost by one of the players between positions,
then the positions are considered to be different. One case where such repetition
typically occurs is during a perpetual check. In the game (Britton vs. Crouch, London
P&D Knights, 1984) Black did ‘check’ his opponent 43 consecutive times and avoided
a draw. White resigned after move 90.
1.3.3 50-Move Rule
If there have been 50 consecutive moves without any piece captured or pawn moved,
then either player may claim a draw. This law has been changed by the FIDE between
50, 75 and 100 moves over the years (Hooper and Whyld, 1996). It was done to
accommodate new research that has shown certain endings e.g. KRB vs. KR to require
more moves to force a win. However, the FIDE has since returned to the original 50-
move rule (FIDE, 2005).
244
1.3.4 Touching Pieces
When a player touches one of his own pieces, he is obliged to make a legal move with
it. If a player touches an opponent’s piece, he is obliged to capture it, if possible.
Adjusting a piece is permitted by first saying “J'adoube” (French for ‘I adjust’).
1.4 Chess Notation
The squares of the chessboard are identified using a coordinate system of letters and
numbers (Figure A.13). The vertical ‘files’ are labelled a through h and the horizontal
‘ranks’ are numbered 1 through 8. Each piece type other than the pawn is identified by
an uppercase letter. The following letters are typically used for the chess pieces: King
(K), queen (Q), rook (R), bishop (B) and knight (N).
Black
White
Figure A.13 The Chessboard and its Coordinates
These may differ between players who speak different languages. Icons are sometimes
used to solve this problem. Pawns are only indicated by their algebraic coordinate. A
245
move is recorded using the letter to denote the piece type and coordinate of the
destination square. For example, Ne4 means to move a knight to the e4 square and f6
simply means to move a pawn to the f6 square. When a piece makes a capture, an ‘x’ is
inserted in the centre of the notation. For example, Nxe4 means the knight has captured
an opponent piece on e4. When a d-file pawn captures say on e6, the notation is written
as dxe6. En passant captures are specified using the destination square, not the one the
opponent pawn is on. It may be followed by ‘e.p.’ to denote the special manoeuvre.
If two or more pieces of the same type can move to the same destination, the piece
alphabet (e.g. K, Q) remains. However, this is followed in sequence (as they differ) by
the original file, the rank or both. For instance, two knights of the same colour on b3
and e4, which can both move to c5, should be written as Nbc5 or Nec5. If the knights
were on g3 and g7, the notations would be N3f5 or N7f5. A capturing manoeuvre in
such situations is indicated by an ‘x’ placed just before the destination square
coordinate.
When a pawn promotes, the chosen piece is indicated after the move, e.g. d8Q or c8B.
Sometimes an equal sign (‘=’) is used e.g. g8=Q. Castling is indicated using 0-0 for the
kingside and 0-0-0 for the queenside. This is the same for both White and Black. A
checking move typically has a ‘+’ sign added to the end of the notation. Checkmate is
indicated using ‘#’ or ‘++’. The notation ‘1-0’ at the end of all the recorded moves
indicates that White won whereas ‘0-1’ means that Black won. A draw is indicated
using ‘½-½’. Chess notation can be written in two parallel columns or in a continuous
line of text (e.g. 1. e4 e5 2. Nf3 Nc6 3. Bc4 Nf6) which saves space.
246
Additionally, the following characters (shorthand notation) might be added after a move
as part of its commentary (Table A.2). Those listed here are the ones most commonly
found.
Table A.2 Shorthand Notation
Character Meaning
! A good move !! A very good move ? A mistake ?? A bad mistake ??? A critical mistake
(possibly resulting in forced mate) !? An interesting but uncertain move ?! A doubtful move
1.4.1 Board Notation
Chess positions can be stored in the form of the widely-used Forsyth-Edwards Notation
(FEN). Using FEN, a complete game position can be recorded in one line of text. It is
based on White’s perspective of the game, like in standard diagrams. The notation starts
from rank 8 (a8 through h8, then to a7 etc.) all the way down to rank 1 (the h1 square).
Each piece is identified by a single letter as used in the previous section. White pieces
use uppercase letters, whereas Black pieces use lower case. Blank squares are noted
using numerals (i.e. 1 through 8) and a slash (‘/’) separates ranks. The characters ‘w’
and ‘b’ then indicate which side is to move.
Castling permissions are indicated using ‘K’ (White can castle on the kingside), ‘Q’
(queenside) and followed by similarly lowercase for Black. An ‘en passant’ target
square (the empty one) is indicated using its algebraic coordinate. If there are no
castling permissions or ‘en passant’ moves possible, the dash (i.e. ‘-’) is used for each.
The number of plies since the last pawn move or capture is noted to determine if a draw
247
can be claimed (50-move rule, see Appendix A, subsection 1.3.3) and finally the
number of the full moves played following White (starts with 1).
The starting position (see Figure A.1) is described in FEN as:
rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1
The position in Figure A.7(a) is written as:
r3kbnr/pppqpppp/2np4/8/2B1P1b1/2N2N2/PPPP1PPP/R1BQK2R w KQkq - 0 1
248
APPENDIX B: GLOSSARY of CHESS TERMS
The following is a list of chess-related terms used in this thesis and their meanings
(Rice, 1997; White, 2003).
1. Anticipation – a chess composition is said to be anticipated when its theme has
already appeared in an earlier problem without the knowledge of the later composer.
The board configuration does not have to be exactly the same, just similar.
2. Castling – a single move in chess involving the king and either of its rooks, all in
their original position. Castling involves moving the king two squares towards a
rook, then moving the rook onto the square next to it (on the other side).
3. Check – a term used to describe when a king is threatened with capture on the next
move.
4. Checkmate – a term used to describe when a king has no escape from capture on
the next move. This means the end of the game.
5. Chess composition – also known as a chess problem, is a sort of puzzle set by
someone (usually referred to as a composer) using chess pieces on a chessboard that
presents the solver with a stipulation or particular task to be achieved (e.g. White to
play and win).
6. Chess variant – a game essentially derived from chess with at least a difference in
board type, pieces or rules (e.g. Chess960, Handicap chess).
7. Combination – several pieces working together to achieve a particular task or goal
(e.g. checkmate). Loosely, it also means a sequence of at least 3 moves.
8. Composition conventions – general rules or guidelines that are typically adhered to
by composers. Collectively, they may be referred to as simply, ‘composition
convention’ (without the s).
249
9. Cook – a chess problem is said to be cooked when there is a second ‘key move’ not
intended by the composer.
10. Direct-mate – a type of chess problem where White is to play and win in n moves,
against any defence.
11. Discovered check - a check incidentally delivered by a piece when a friendly one
moves out of its line of attack. This is also a common theme in chess.
12. Dual – when the solution to a composition presents White with more than one valid
continuation (after the ‘key move’). The terms ‘triple’ and ‘multiple’ are used when
there are more than two such continuations.
13. Elo rating – a method for calculating the relative skill level of a chess player,
created by Arpad Elo, an American physics professor. It was adopted by the FIDE in
1970.
14. En passant – a move in chess (French for ‘in passing’) where a pawn can capture an
opponent’s pawn immediately after it moves two squares forward from its starting
position, as if it had only moved one square forward.
15. En prise – a term used to describe when a piece (usually undefended) is put in a
position to be captured by the opponent.
16. Endgame study – a type of chess problem that is more likely to occur in a real
game and has a stipulation that is less specific (e.g. ‘White to play and win’) than
direct-mate problems and takes more moves to accomplish.
17. FEN - Forsyth-Edwards Notation. A method of describing a complete chess
position in a single ‘string’ or line of text.
18. FIDE - Fédération Internationale des Échecs or World Chess Federation. An
international organization that acts as the governing body of international chess. It
was founded in Paris, France in 1924.
250
19. File – a vertical line of 8 squares on the chessboard (usually identified as a through
h in chess notation).
20. Flight square – a square onto which a piece, usually the king, can move to if it is
threatened with capture.
21. FM – short for FIDE Master (typical Elo rating ≥ 2300), which ranks below IM
(International Master).
22. GM – short for Grandmaster (typical Elo rating ≥ 2500). This is the highest title or
rank a chess player can attain short of ‘World Champion’.
23. Helpmate – a type of chess problem where White and Black work together to put
the latter in checkmate within a specific number of moves. Black usually moves
first.
24. Idea – a specific implementation of a theme. An idea can be plagiarized whereas a
theme cannot.
25. IM – short for International Master (typical Elo rating between 2400 and 2500).
26. Key – the first move in the solution of a chess problem.
27. Meredith – a chess composition containing between 8 and 12 pieces.
28. Model mate – a ‘pure mate’ in which all the white pieces are involved. Possible
exceptions include the king and pawns.
29. NN – ‘Nescio Nomen’ which is Latin for ‘name unknown’. This is sometimes used
in the notation of a chess game in place of a player’s name (when it is unknown).
30. Opening book – a database of openings provided to chess programs that eliminates
the need for them to calculate the best lines during the first 10-12 moves of the game
where the positions are difficult to evaluate.
251
31. Orthodox problems – compositions restricted to pieces that are used in the official
game. ‘Fairy chess’ for example, is not considered orthodox because it uses
additional pieces with special moves that are not present in the standard,
international version of the game.
32. Passed Pawn – A pawn (typically in the endgame) that has no opponent pawns on
the same or the adjacent files to prevent it from promoting.
33. Perpetual Check – This occurs when one player is faced by an unending series of
checks by the opponent. The result is usually a draw.
34. PGN – Portable Game Notation. A popular and accessible computer file format for
recording chess games.
35. Piece field – one square in every direction immediately around a piece, including its
own.
36. Ply – a half-move of either White or Black (e.g. Nd5). A full move is two ply.
37. Promotion – when a pawn reaches the last rank and can be exchanged for a queen,
rook, knight or bishop.
38. Pure mate – a checkmate position in which the checkmated king and all vacant
squares in its immediate field (i.e. the squares around it and the one it is on) are
attacked only once. Squares in the king's field occupied by its friendly pieces must
not also be attacked by the winning side unless such a piece is necessarily pinned to
the king.
39. Rank – a horizontal line of 8 squares on the chessboard (usually identified as 1
through 8 in chess notation).
40. Retrograde Analysis – a method used by chess problem solvers to determine which
moves lead up to a particular position.
41. Selfmate – a type of chess problem where White forces Black to win (i.e.
checkmate) within a specific number of moves.
252
42. Stalemate - a position where the player whose turn it is has no legal moves and is
also not in check. The result is a draw.
43. Tempo – a single move (i.e. turn) gained or lost by accomplishing a task in one
fewer or more moves, respectively. A tempo can also be gained by causing the
opponent to lose one. For example, a piece capable of moving to a desired square in
one move but instead takes two loses a tempo.
44. Theme – the strategic motive of a move combination (e.g. fork, pin).
45. Try – a move in a chess problem which appears to solve it, but is actually not the
correct one (i.e. the ‘key move’) because there is one defence by Black against it.
46. Triple – see ‘dual’.
47. Variant – see ‘chess variant’.
48. Variation – an alternative line of play in the solution to a problem or combination
in a real game.
253
APPENDIX C: EXAMPLE POSITIONS
The following positions are provided to illustrate examples referenced in this thesis.
Included is the relevant location (section, subsection or Appendix) where each position
is cited and additional commentary where appropriate. To understand the chess notation
used, see Appendix A, section 1.4.
Subsection 3.3.1
XABCDEFGHY
8k+N+-+-+(
7zP-+p+-+-'
6-+-+p+-+&
5+-zp-+-+-%
4-+-+-+-vl$
3+-+-+-+-#
2-+-+-+-+"
1+-+-+-wQK!
xabcdefghy
J. Møller, Skakbladet 1920, Mate in 3
Figure C.1 A Typical ‘Logical’ School Composition
White’s main plan is 1. Qb1, threatening 2. Qb8# but is refuted by 1. … Bg3. The key
move is 1. Qg7! which threatens 2. Qxd7 followed by inevitable checkmate. Black can
defend by 1. … Be7. This signifies the success of White’s foreplan which was to decoy
the bishop. Thus White can now revert to the main plan with 2. Qb2, again threatening
3. Qb8#. Black can no longer defend with 2. … Bg3 but can play 2. … Bd6, allowing 3.
Qg2# (Levitt and Friedgood, 2008).
254
Subsection 3.3.3
XABCDEFGHY
8-+-+-+-+(
7+-+-+-+-'
6-+-+-+-+&
5+-+-+-+-%
4k+-+-+-+$
3+-tR-+-+-#
2p+-+-+N+"
1+-+-+-+K!
xabcdefghy
Figure C.2 J. Mintz, The Problemist, 1982, Helpmate in 3 (Black to Play)
The solution is: 1. a1=Q+ Ne1 2. Qb2 Nc2 3. Qb5 Ra3# (Feather, 2008).
XABCDEFGHY
8rsnl+k+ntr(
7zp-+p+pzpp'
6-+-+-+-+&
5+pvlN+N+P%
4-+-+PvLP+$
3+-+P+Q+-#
2PwqP+-+-+"
1tR-+-+KtR-!
xabcdefghy
Adolf Anderssen vs. Lionel Kieseritzky, London, 1851
Figure C.3 The ‘Immortal Game’ (after 17. … Qxb2)
Here, Anderssen plays 18. Bd6 and leaves both his rook on a2 and g1 ‘en prise’. The
game continues with 18. … Bxg1 19. e5 Qxa1+ 20. Ke2 Na6 21. Nxg7+ Kd8 22. Qf6+
Nxf6 23. Be7#. A ‘spectacular’ sacrifice of a queen and two rooks. Decades later,
analysts have found that 18. d4, 18. Be3 and 18. Re1 were also wins without the need to
sacrifice as much material (Kasparov, 2003). However, this game is considered
255
beautiful and ‘immortal’ because of the actual line played and sacrifices that ensued
(Shenk, 2006).
Subsection 3.5.1
XABCDEFGHY
8rsnlwqr+k+(
7zpp+-+pzpp'
6-+pvl-sn-+&
5+-+p+-+-%
4-+-zP-+-+$
3zP-sNLzP-+-#
2-zPQ+NzPPzP"
1tR-vL-+RmK-!
xabcdefghy
Figure C.4 Kasparov vs. Deep Junior, Game 5, New York, 2003
At this point, the computer ‘Deep Junior’ plays the unexpected move 10. … Bxh2+,
which commentators thought was a very unlikely move for a computer to make (Ban,
2004). Kasparov accepted the sacrifice with 11. Kxh2 and exposed his king. The game
ended in a draw on move 19. Such ‘positional’ sacrifices are rare in computer chess.
XABCDEFGHY
8r+lwqkvl-tr(
7zpp+n+pzp-'
6-+p+psn-zp&
5+-+-+-sN-%
4-+-zP-+-+$
3+-+L+N+-#
2PzPP+-zPPzP"
1tR-vLQmK-+R!
xabcdefghy
Figure C.5 Deep Blue vs. Kasparov, Game 6, New York, 1997
256
Here, the IBM computer program plays 8. Nxe6 which gives White a good positional
advantage. As part of the program’s opening book theory, the sacrifice is not as
dramatic as it would have been had Deep Blue arrived at the move on its own. The
game continued with 8. … Qe7 9. 0-0 fxe6 10. Bg6+. Kasparov resigned on move 19.
Section 4.2
XABCDEFGHY
8-+-+-+-+(
7+Q+-+-+-'
6-+-mk-mK-+&
5+-zpP+-+-%
4-+-+-+-+$
3vL-+-+-+-#
2-+-+-+-+"
1+-+R+-+-!
xabcdefghy
Figure C.6 Two-way Discovered Checkmate
In the constructed position above (Figure C.6), Black has just moved his pawn from c7
to c5 to intervene against the bishop on a3. White can now capture the pawn ‘en
passant’ with a two-way discovered checkmate by playing dxc6#. Based on the same
concept, if the king was not checkmated or another piece was the target, it would be a
two-way discovered check or attack, respectively.
257
Subsection 4.5.4
XABCDEFGHY
8K+-+R+-+(
7+Q+N+-+-'
6-+-zpPzp-+&
5+-zpk+-sn-%
4-+pvlnzp-+$
3+-+-zpltr-#
2-+-zp-tr-+"
1+-+-+-+-!
xabcdefghy
Gerald Frank Anderson, Western Morning New, 1922
Figure C.7 Two-Phase Piece Removal (Economy)
In Figure C.7, the checkmate, when reduced economically using one phase of piece
removal, is left without the pawn on e6. This makes the knight on d7 (which was tested
only prior to the pawn’s removal) irrelevant or passive. Hence a second phase,
preferably starting with the weakest piece on the board, is also necessary.
Section 5.1
XABCDEFGHY
8-+l+-+-tr(
7zp-+-+-+-'
6-+-+-+p+&
5+P+pmkpzP-%
4-+-tR-tR-zP$
3+-+Ltr-+-#
2-+P+-+K+"
1+-+-+-+-!
xabcdefghy
NN vs. Mannheimer, Frankfurt am Main, 1921
Figure C.8 Activated Fork
258
In Figure C.8, 1. ... Re4 activates the fork by the black king on both white rooks even
though neither one was threatened just prior. The move simply cuts off the white rooks
from their defence of each other. Such ‘activated’ forks are not very common. Based on
the proposed aesthetics model, a single move like this would not score for its ‘intended’
purpose (see section 2.1). However, it could be compensated by other aesthetic features
of the combination. The move played in this position for example (i.e. Re4), does
register as a heuristic violation because it is left ‘en prise’ to the bishop on d3.
XABCDEFGHY
8-+-+k+-+(
7+-+-+-+-'
6-snn+-+-+&
5+Q+-+-+-%
4-+-+-+-+$
3+-+-+-+-#
2-+-+-+-+"
1mK-+-+-+-!
xabcdefghy
Figure C.9 Repeated Fork
In the constructed position shown in Figure C.9, both knights are already forked by the
queen (the one on c6 also happens to be pinned). The move Qc5, repeats the fork
without penalty. Technically, it is not identical to the previous fork.
259
Section 5.2
XABCDEFGHY
8-+-vlksn-+(
7mKP+-+-+-'
6-sN-+q+P+&
5+-+-+-+-%
4-+L+-+-vL$
3+-+-+-+-#
2-+-+ +-+"
1+-+-tR-+-!
xabcdefghy
Figure C.10 Immobilizing the Queen with a Two-way Pin
In the constructed position shown in Figure C.10, White has just played Bc4! The queen
is completely immobilized because it is now pinned both against its king on e8 and the
mating square on f7. This can also be considered a ‘discovered’ pin (i.e. the rook on e1
being the discovered pinning piece).
XABCDEFGHY
8k+-+-+r+(
7+q+-+n+-'
6-+-+-+-+&
5+-+p+-+-%
4-+-+-+-+$
3+n+-+L+-#
2r+-+-+-+"
1+-+R+K+-!
xabcdefghy
Figure C.11 A Three-way Pin (Bxd5)
260
XABCDEFGHY
8-+K+-+-+(
7+-+-vL-+-'
6-zp-+-+-+&
5+-+-mkpzp-%
4-+-sN-sN-+$
3zP-+p+-+-#
2rvl-+-+Q+"
1+-+-+-+-!
xabcdefghy
Williams, P. H., Reading Observer, year unknown, Mate-in-3
Figure C.12 Negative Evaluation for the Pin after 1. Qg2
Section 5.4
XABCDEFGHY
8K+-+-+-vL(
7vL-+-+-wq-'
6-+-+-+-+&
5+-wq-+-+-%
4p+-sn-+-+$
3mk-zP-+-+-#
2-+-+-+-+"
1+-+-+-vL-!
xabcdefghy
Figure C.13 A Double X-ray with 1. Bxd4
A double x-ray is highly uncommon. There are two reasons for this. First, it is difficult
to arrive at a configuration where at least one of the x-rays was not already in effect
(hence the knight on d4; without it there would be no x-ray along the a7-d4 diagonal).
Second, the moving piece cannot be one that the opponent can capture favourably. This
means that if a white queen, instead of a bishop, was on g1, and captured the knight on
261
d4, there would be no valid x-ray at all; because from that point, either queen could
force an equal exchange on d4 with impunity. So this sort of x-ray is essentially limited
in the types of pieces that could be involved. Not to mention that in the given position,
there happens to be three dark-square bishops.
Section 5.5
XABCDEFGHY
8-+-+-+-wQ(
7+l+-+-+-'
6-+-zpP+-+&
5+-+-+-+-%
4-+N+-+-+$
3+-+-+-+-#
2rzp-+-+-+"
1mk-+-mK-+R!
xabcdefghy
Bottger, H., Die Schwalbe, 2007
Figure C.14 Castling as a Discovered Attack Manoeuvre (0-0#)
XABCDEFGHY
8-+-mk-+-+(
7+-+-+-+-'
6-+-+-+r+&
5+-+-+-+-%
4-+-+-sN-+$
3+-+-+-+-#
2-+-+-+-+"
1+LmKR+-+-!
xabcdefghy
Figure C.15 Double-Discovered Attack (after 1. Ndf4+)
262
In this constructed position, the knight has just moved from d3 to f4. It creates a double
attack on the rook at g6 (knight + bishop) and another double attack on the king at d8
and (again) the rook at g6. CHESTHETICA (see Appendix D) was not explicitly
programmed to handle multiple occurrences of the discovered/double attack theme in a
single move because even a single occurrence has many permutations and conditions.
Even so, the program had apparently placed emphasis on the double attack involving the
king and rook (full evaluation), and evaluated the double attack involving the knight and
bishop as a mere discovered attack on the rook at g6 by the bishop on b1. The total
score was therefore 0.788 + 0.269 = 1.057.
The author tested a few other configurations and learned that the program would place
emphasis (i.e. full evaluation of both pieces) on an attack involving the king regardless
of piece order (i.e. upper left of the board to lower right) and partial emphasis (i.e. the
discovered piece only) on the second attacking pair. This tended to keep the aesthetic
score for the theme from getting too high. If the king is not involved, emphasis is placed
on the pair that (at least one piece of which) attacks the enemy piece first in order. Cases
like these are not very common in the theme, and it is difficult to anticipate and hard-
code them into a computer program. However, if they do occur, the above method or
something similar may be adopted.
263
D (Refer Appendix)
XABCDEFGHY
8-snl+-vlntr(
7+Q+-zpkzp-'
6-+-+-zp-tr&
5+-+-+-+p%
4-+P+-+-zP$
3+-+q+-+-#
2PzP-zPPzPP+"
1tRNvL-mKLsNR!
xabcdefghy
Upmark vs. Johansson, SWE-ch U18m, 1995
Figure C.16 Stalemate in 3
The game lasted only 10 moves. It may have been planned in advance by both players.
The last three moves - while not meeting the desired objective of checkmate – scores
0.697 aesthetically.
1. c4 h5 2. h4 a5 3. Qa4 Ra6 4. Qxa5 Rah6 5. Qxc7 f6 6. Qxd7+ Kf7 7. Qxb7 Qd3
8. Qxb8 Qh7 9. Qxc8 Kg6 10. Qe6 ½-½
264
APPENDIX D: CHESTHETICA
A computer program called, CHESTHETICA, was developed by the author to detect
and evaluate the aesthetic principles and themes as described in chapters 4 and 5.
Manual evaluation of these principles was not feasible because the process is tedious
and prone to error. The computer program made evaluation of tens of thousands of
chess combinations possible. It was developed over the last 18 months of the research
period using Microsoft Visual Basic 6. The author’s prior experience with this
programming language was the main reason it was chosen. In principle, any
programming language could have been used. Initial programming work, new features,
enhancements and debugging resulted in a total of 531 ‘builds’ to the latest version.
The program contains approximately 12,000 lines of code or the equivalent of over 200
A4 pages in print (which is why the source code was not included here). It was designed
specifically for the scope of analysis (i.e. mate-in-3 combinations) with over 130
specialized functions and subroutines to handle the various tasks required for the
research. Even so, the program is also capable of processing helpmates and stalemates
in 3. A curious example of the latter – even though it does not meet the desired
objective of checkmate - occurred in the game (Upmark vs. Johansson, SWE-ch U18m,
1995). The last three moves of the game from the position (FEN:
1nb2bnr/1Q2pkp1/5p1r/7p/2P4P/3q4/PP1PPPP1/RNB1KBNR w KQ) score 0.697
aesthetically (see Appendix C, Figure C.16).
CHESTHETICA has no game-playing intelligence per se but is capable of facilitating a
full chess game (including all the special rules, e.g. en passant, castling, promotion)
between two players. This was a necessary foundation for automated aesthetic analysis.
265
The program, while not explicitly coded with performance in mind, is capable of
analyzing an average of 3.7 combinations per second on a Pentium 4, 3.2 GHz machine
with 2 GB of RAM running Windows XP Service Pack 3.
Combinations from tournament games are analyzed at a slower rate (2.5 per second)
because the whole game needs to be played until the end, reversed three moves and only
then analyzed (to capture the starting position of the mate-in-3 combination).
Screenshots of the program are shown below. Figure D.1 shows the main interface and
Figure D.2 the ‘About Box’.
Figure D.1 The Main Interface to CHESTHETICA
266
Figure D.2 The ‘About Box’
Game databases can be imported into CHESTHETICA in the form of PGN files with
annotations (if any) removed. However, since the program is designed for mate-in-3
combinations, the games must end with checkmate. Direct-mate compositions
inherently do end with mate, so there are no compatibility problems unless they are
longer or shorter than 3 moves. The program can analyze individual combinations or be
set to analyze the whole database automatically while keeping track of the scores and
other relevant details. The aesthetics evaluation panel is shown in Figure D.3.
This panel displays the individual scores for each aesthetic principle and theme in the
current combination (the analysis shown in Figure D.3 is not related to the position in
Figure D.1). Automatic commentary is generated as each point of evaluation (see
section 3.8) is reached to explain relevant details about how those scores were
calculated. This made it easier to understand the component scores behind combinations
of interest. Each aesthetic principle and theme can also be analyzed on demand (per
move) using the button at the top of each column (with the name of the principle/theme
on it). The panel also shows the independent cumulative scores of the aesthetic
principles and themes.
267
Figure D.3 The Aesthetics Evaluation Panel
Figure D.4 shows a bar chart that the program can automatically generate upon
completing the analysis of a database of combinations. The chart shows the number of
theme instances that occurred in the collection (this feature was used in Experiment 1,
see section 6.1). The table to the right keeps track of each theme and the combination in
which it appeared. Selecting such a combination causes it to appear automatically in the
main window (see Figure D.1). This chart proved useful in determining if a particular
theme existed in a database and how often it occurred. Some themes, like the x-ray, are
quite rare and difficult to locate manually. A similar chart was not implemented for the
aesthetic principles because only three of the seven can vary in frequency (see
subsection 6.1.2). This information can be exported to a text file.
268
Figure D.4 The Thematic Frequency Chart
The aesthetic evaluation scores are written to an ASCII text file in the program folder
automatically (in case of power failure etc.). To speed up certain experiments, analysis
of specific principles and themes was necessary. Figure D.5 shows the interface which
enabled such a selection.
Figure D.5 Aesthetic Principle and Theme Selection
269
The pawn with a red cross over it presents the option of excluding pawns as possible
threats for the fork, pin, skewer and ‘discovered/double attack’ themes. This selection
feature also helped to a large extent with debugging because individual functions could
be disabled systematically to help identify the source of an error. A checkmate solver
(Figure D.6) was implemented using the ‘minimax’ algorithm with alpha-beta pruning.
This was done out of curiosity while studying it.
Figure D.6 The Mate Solver
There was no intention to use the mate solver for this research because aesthetic
analysis did not require seeking for forced mates. However, for the surveys (see section
6.6), it proved to be extremely useful in identifying combinations from the data sets (see
section 6.0) that were forced lines of play. While it may seem easy, in principle, to
‘link’ to an external (and perhaps more efficient) mate solver program, a suitable one
could not be found. This is because every combination in the tournament game data set,
for example, had to be tested in terms of being an exclusive mate-in-3 combination.
This means that the main line of the combination had to be a forced one and that mate
was not possible in any more or any fewer than 3 moves. Many mate solvers simply find
270
the shortest checkmate possible in a given position (and possibly other variations of it)
but lack the ability of looking for mates of only a specific length. The mate solver in
CHESTHETICA, while likely slower than other solvers, was therefore useful. It could
be programmed exactly as required. The 12,552 tournament game combinations took
several days of processing on 5 different computers before the exclusive mate-in-3
combinations were identified (see section 6.6). The following is an actual code snippet
taken from CHESTHETICA.
Private Function Same_Line(ByVal point1 As Integer, ByVal point2 As Integer, rfd As String) As Boolean 'determines if two squares are on the same rank, file or diagonal Dim point1_rank As Integer Dim point1_file As Integer Dim point2_rank As Integer Dim point2_file As Integer Dim file_diff As Integer Dim rank_diff As Integer If Within_Board(point1) = False Or Within_Board(point2) = False Then Exit Function point1_file = (point1 Mod 8) + 1: point2_file = (point2 Mod 8) + 1 point1_rank = 8 - (point1 \ 8): point2_rank = 8 - (point2 \ 8) file_diff = Abs(point2_file - point1_file): rank_diff = Abs(point2_rank - point1_rank) Select Case rfd Case "rank" If point1_rank = point2_rank Then Same_Line = True Case "file" If point1_file = point2_file Then Same_Line = True Case "diagonal" If rank_diff = file_diff Then Same_Line = True Case "any" If (point1_rank = point2_rank) Or (point1_file = point2_file) Or (rank_diff = file_diff) _ Then Same_Line = True End Select End Function
A Microsoft Windows (98, XP, Vista) installer package file of the full program is
available to other researchers upon request from the author.
271
APPENDIX E: PSEUDOCODE
The following are ten blocks of pseudocode for a selection of the evaluation functions
proposed in this thesis (those not included here were deemed easy enough to understand
from the explanations provided in the main text). Their purpose is to aid comprehension
and simplify the process of codifying the evaluation functions (for those who wish to
replicate or improve upon the experimental results). Commentary is given in italics.
Included also is the relevant location (section, subsection) where the function is
explained in the main text.
Subsection 4.1.2
Violate Heuristics Successfully: Capture Enemy Material
Declare cvt /*capture value threshold Declare udmv /*value of undefended material if capturing_move(moving_piece) = TRUE then cvt = piece_value(captured_piece(moving_piece)) end if /*if the previous move captured a piece, the violation is only for pieces of higher value than the captured one undo_move() for i = 0 to 63 if piece_color(i) ≠ "black" then skip if piece_type(i) = "pawn" then skip if piece_value(i) ≤ cvt then skip if favourable_capture(i, “by white”) = TRUE then udmv = udmv + (piece_value(i)/9) end if next i redo_move() Return udmv
272
Subsection 4.1.3
Violate Heuristics Successfully: Do Not Leave Your Own Pieces ‘En prise’
Section 4.3
Use All of the Piece’s Power
Declare dmop /*distance travelled by moving piece Declare dmat /*distance travelled by mating piece Declare apps /*principle score Declare mating_piece Declare black_king dmop = distance(location(moving_piece), destination(moving_piece)) apps = dmop/piece_power(moving_piece) if checkmated(“black”) = TRUE then mating_piece = determine_mating_piece() black_king = location(“black king”) dmat = distance(location(mating_piece), black king) apps = apps + (dmat/piece_power(mating_piece)) end if /*if Black is also checkmated (i.e. the final move), the principle score is incremented accordingly Return apps
Declare cvt /*capture value threshold Declare empv /*value of en prise material undo_move() if capturing_move(moving_piece) = TRUE then cvt = piece_value(captured_piece(moving_piece)) end if /*if the previous move captured a piece, the violation is only for en prise pieces of higher value for i = 0 to 63 if piece_color(i) ≠ "white" then skip if piece_type(i) = "pawn" then skip if piece_value(i) ≤ cvt then skip if favourable_capture(i, “by black”) = TRUE then empv = empv + (piece_value(i)/9) end if next i redo_move() Return epmv
273
Section 4.5
Checkmate Economically
Declare fp[] /*friendly piece array Declare ppmf /*passive piece maximum fields Declare acf /*active control fields Declare ols /*overlapping squares Declare ap[] /*active piece array Declare ekd[] /*enemy king domain array Declare phc /*phase count Declare fpc /*friendly piece count backup_board() for phc = 1 to 2
if phc = 1 then fp[] = locate_and_sort_pieces(“white”, descend) fpc = upper_bound(fp)
else fp[] = locate_and_sort_pieces(“white”, ascend)
end if
for i = 0 to upper_bound(fp) remove_from_board(fp[i]) if checkmate(“black”) = FALSE then
replace(fp[i]) : ap(j) = fp[i] : j = j + 1 else remove(fp[i]) : update_board() ppmf = ppmf + max_field(fp[i]) end if next i
next phc ekd[] = generate_king_domain(“black king”) for k = 0 to upper_bound(ap)
acf = acf + active_field(ap[k], ekd[]) / max_field (ap[k]) next k ols = overlapping_squares(ap[], ekd[]) : restore_board() Return [acf – ((ols + ppmf) / 9)] / fpc
274
Section 4.7
Spread Out the Pieces (Sparsity)
Section 5.1
The Fork
Declare fc = 37 /*fork theme constant Declare fpp /*forking piece power Declare fpms[] /*forked pieces and mating squares array Declare vfpms[] /*forked piece or ms value array Declare msq[] /*mating square(s) array Declare disrto /*distance ratio Declare pcm /*possible checking moves Declare legal_moves[] /*legal moves array /*loc = location on the board /*intv = intervening piece(s) if favourable_capture(loc(forking_piece), “by black”) = TRUE then
exit function end if /*if the (tentative) forking piece can be captured favourably by the opponent, the theme is invalid /*the following block scans the board for forked pieces and mating squares legal_moves[] = generate_legal_moves(forking _piece) /*stored in order from the upper left to lower right of the board
Declare pc as Integer /*piece count Declare sp as Integer /*surrounding pieces Declare Const board_size = 63 /*e.g. chess, checkers for i = 0 to board_size if current_position(i) <> 0 Then pc = pc + 1 sp = sp + count(pieces_in_field(i)) end if next i If pc = 0 then exit function /*empty board? Return 1 / [(sp/pc)+1]
275
for i=0 to upper_bound(legal_moves) if piece_type(i) = “pawn” or piece_color(i) = “white” then skip /*squares with pawns and white pieces are not evaluated if piece_color(i) = “black” then if favourable_capture(i, loc(forking_piece)) = TRUE then /*the forking piece can capture it favourably fpms[j]=i /*stores its location if mating_square(i) = TRUE then vfpms[j] = piece_value(“king”) msq[k] = i /*a record of the mating square locations k=k+1 else vfpms[j]=piece_value(piece_type(i)) end if j=j+1 end if end if if piece_type(i) = “empty” then if mating_square(i) = TRUE then fpms[j] = i vfpms[j] = piece_value(“king”) msq[k] = i j=j+1 : k=k+1 end if end if next i if upper_bound(fpms)<2 then exit function /*a fork must have at least two pieces or mating squares /*the following block removes invalid mating squares (e.g. one of two on the same line with no intervening pieces between) for l=0 to upper_bound(fpms) If piece_type(fpms[l]) = “king” then remove_mating_square(fpms[],vfpm[]) exit for /*if the enemy king is forked, no mating squares count end if if mating_square(fpms[l]) = TRUE then for m=0 to upper_bound(msq) /*each mating square is tested against all the others If same_line(fpms[l], msq[m]) = TRUE then if intv(fpms[l], msq[m])= FALSE then remove_mating_square(fpms[l],vfpm[l]) end if end if next m end if next l if upper_bound(fpms)<2 then exit function fpp = piece_power(piece_type(forking_piece)) /*the final block below evaluates the legitimately forked pieces and mating squares
276
for n = 0 to upper_bound(fpms) /*invalid mating squares have now been removed tfpv = tfpv + vfpms[n] disrto = disrto + [distance(loc(forking_piece),fpms[n])/fpp]
/*the distance to piece power ratio between the forking and _ forked piece
pcm = pcm + possible_checking_moves(fpms[n]) next n /*the number of prongs is equivalent to n Return [1/fc * (tfpv + n + disrto – pcm)]
Section 5.2
The Pin
Declare pc = 20 /*pin theme constant Declare ppp /*pinning piece power Declare pins[] /*pinned piece array Declare vpins[] /*pinned piece value array Declare target[] /*target piece array Declare vtarget[] /*value of target piece array Declare legal_moves[] /*legal moves array Declare mov_sq /*movable squares of the pinned piece Declare intrv_scr /*score for intervening pieces Declare chk_mvs /*possible ‘checking’ moves Declare disrto /*distance ratio Declare prtl_pwr /*partial pin power Declate cpav /*cumulative pin aesthetic value /*loc = location on the board if favourable_capture(loc(pinning_piece), “by black”) = TRUE then
exit function end if /*if the (tentative) pinning piece can be captured favourably by the opponent, the theme is invalid /*also, to enter this function, only a long-range piece would qualify legal_moves[] = generate_legal_moves(loc(pinning_piece)) /*stored in order from the upper left to lower right of the board for i=0 to upper_bound(legal_moves) if piece_type(i) = “pawn” or piece_color(i) = “white” then skip /*pawns and white pieces are not evaluated if pinned(i, loc(pinning_piece)) = TRUE then /*this tests for black pieces that are pinned
/*the pinned() function also ensures that the pin is favourable undo_move() if alrdy_pinned(i, loc(pinning_piece)) = FALSE then /*if the piece was not already pinned
277
redo_move() pins[j] = i /*locations of the pins, i.e. pinned pieces vpins[j] = piece_value(i) target[j] = target_piece(i, pinning_piece) /*locations of the corresponding target pieces _ that might include mating squares vtarget[j] = piece_value(target[k]) if mating_square(target[k]) = TRUE then /*this includes mating squares with pieces on them vtarget[j] = piece_value(“king”) /*if so, it is valued equivalent to the king end if j=j+1 else redo_move() end if end if next i for k = 0 to upper_bound(pins) /*all the pinned and target pieces (if any) must be accounted for
ppp = piece_power(piece_type(pinning_piece)) mov_sq = upper_bound(pinning_line(pins[k],target[k], _ loc(pinning_piece))) /*the number of movable squares the pinned piece has along the _ pinning line prtl_pwr = mov_sq/piece_power(piece_type(pins[k])) /*partial pin power (based on its movable squares) intrv_scr = (intv_scr(pins[k], loc(pinning_piece)) /*the intv_scr() function calculates the score for all possible and valid intervening pieces; this is then attributed to the variable intrv_scr chk_mvs = upper_bound(possible_ checking_moves(pins[k]), target[k])) /*the number of possible checking moves by the pinned piece _ and target piece
disrto = disrto + [distance(loc(pinning_piece),target[i])/ppp] /*the distance to piece power ratio between the pinning and _ target piece cpav = cpav + [((vpins[k] + vtarget[k] + disrto) – _ prtl_pwr - chk_mvs – intrv))/pc]
next l Return cpav
278
Section 5.4
The X-Ray
Declare xc = 7 /*x-ray theme constant Declare xr1pp /*x-rayer1 piece power Declare xr2pp /*x-rayer2 piece power Declare xrays[] /*xrayed piece array Declare xrayers2[] /*x_rayer2 piece array Declare vxrays[] /*xrayed piece value array Declare avgxpv /*average x-rayer piece value Declare legal_moves[] /*legal moves array Declare disrto /*distance ratio Declare cxav /*cumulative xray aesthetic value /*loc = location on the board /*to enter this function, only a long-range piece would qualify legal_moves[] = generate_legal_moves(loc(xrayer1_piece)) /*these are the legal moves of the long-range piece that just _ moved; it potentially x-rays an opponent piece for i=0 to upper_bound(legal_moves) if piece_type(i) <> “queen”, “rook” or “bishop” then skip if piece_color(i) = “white” then skip /*only long-range black pieces can be x-rayed if xrayed(i, loc(xrayer1_piece)) = TRUE then
/*the xrayed() function ensures that the x-ray is valid and _ favourable
undo_move() if alrdy_xrayed(i, loc(xrayer1_piece)) = FALSE then /*if the x-rayed piece was not already x-rayed redo_move() xrays[j] = i vxrays[j] = piece_value(i)
xrayers2[j] = xrayer2_id(i, loc(xrayer1_piece)) /*gets the location of the friendly piece that _ completes the x-ray
j=j+1 else redo_move() end if end if next i xr1pp = piece_power(xrayer1_piece) for k = 0 to upper_bound(xrays)
xr2pp = piece_power(xrayers2[k]) disrto = [distance(loc(xrayer1_piece), xrayers2[k]]/ _ min(xr1pp, xr2pp) /*the distance to piece power ratio is the distance between the x-rayer pieces divided by the least powerful (value) of the two
avgxpv = Avg(piece_value(xrayer1_piece), _ piece_value(xrayers2[k]))
279
cxav = cxav + [(1 + Abs(vxrays[k] – avgxpv) + disrto)/xc)] next k Return cxav
Section 5.5
The Discovered/Double Attack
Declare ddac = 20 /*dda theme constant Declare db_atkp[] /*double attack piece array Declare chk_mvs /*possible ‘checking’ moves Declare disrto /*distance ratio Declare dda_score /*theme score Declare legal_moves[] /*legal moves array /*loc = location on the board if favourable_capture(loc(dda_piece), “by black”) = TRUE then
exit function end if /*if the (tentative) ‘dda’ piece (i.e. the one that just moved) can be captured favourably by the opponent, the theme is invalid legal_moves[] = generate_legal_moves(dda_piece) /*stored in order from the upper left to lower right of the board for i=0 to 63 if piece_type(i) = “pawn” or piece_color(i) = “white” then skip /*squares with pawns and white pieces are not evaluated if piece_color(i) = “black” then if disc_attack(loc(dda_piece, “prev”), i) = TRUE then
/*if there is a discovered attack (from any angle)on _ this black piece by a friendly one that was behind the _ one that just moved
discovered_piece = dp_id(loc(dda_piece, “prev”), i)
/*identifies said piece (i.e. the one attacking _ location i) dap = i /*identifies the attacked piece chk_mvs = upper_bound(possible_checking_ moves(loc(dap)) /*the number of possible checking moves by the _ ‘dap’ disrto = distance((loc(dap), _ loc(discovered_piece))/piece_power(discovered_piece) /*the distance to piece power ratio between the discovered piece and the one it attacks
end if
280
if discovered_piece ≠ NULL then for j = 0 to upper_bound(legal_moves) /*legal moves of the moving piece if dbl_attack (loc(dda_piece), j) = TRUE then db_atk_piece = da_id(loc(dda_piece), j) db_atkp[k] = db_atk_piece : k=k+1
/*keeps track of the pieces attacked _ by the moving one
end if next j
/*this tests if the discovered attack is also a double attack
else exit function
/*if there is no discovered attack (at least), the theme is not in effect
end if if upper_bound(db_atkp)>0 then /*if there is a double attack ch_db_atk_piece = max(piece_value(db_atkp))
/*if there is more than one piece, only the _ highest is chosen
chk_mvs = chk_mvs + upper_bound(possible_ checking_moves(loc(ch_db_atk_piece)) /*the number of possible checking moves by the _ piece that is attacked by the moving one
disrto = distance(loc(ch_db_atk_piece), _ loc(dda_piece))/piece_power(dda_piece) /*the distance to piece power ratio between the moving piece and the (chosen) one it attacks
end if if ch_db_atk_piece = NULL then if intv(loc(discovered_piece), dap)= TRUE then /*intervening pieces apply only to discovered _ attacks (not double attacks)
intrv_scr = intv_scr(loc(discovered_piece), _ dap) /*the intv_scr() function calculates the _ score for all the valid intervening pieces; _ this is then attributed to the variable _ intrv_scr
end if end if end if
dda_score = dda_score + [((piece_value(dap) + _ piece_value(ch_db_atk_piece) + disrto) - chk_mvs – _ intrv_scr)/ddac] /*in principle, this calculates the aesthetic score for each _ discovered or double attack created by the moving piece and _ sums them /*note that multiple such instances of the theme are rare and _ certain measures may be required to assess them properly (see _ Appendix C, Figure C.15)
Return dda_score
281
Section 5.6
The Zugzwang
declare legal_moves[] /*legal moves array declare nlm /*number of legal moves declare zc = 30 /*zugzwang constant /*White has just made his move legal_moves() = generate_legal_moves(board_position[]) nlm = upper_bound(legal_moves) swap_turn() /*assume null move by Black if in_check(“black”) = false then /*Black cannot skip his turn if under check
swap_turn() /*restores black’s turn for each legal_moves() make_move() if test_criterion(checkmate) = false then undo_move() : return 0 : exit function end if undo_move() next
else swap_turn() : exit function end if Return nlm/zc
282
APPENDIX F: SURVEY DATA
1.0 Overview of the Surveys
The surveys used for this research were designed by the author and improved based on
feedback from several expert players (i.e. GM Georg Meier, GM Yuri Vovk, IM
Romain Edouard, IM Reiss Tibor and FM Levan Bregadze). They were contacted live
via the Playchess.com network (for registered members). Efforts were taken in the
design to elicit as much cooperation and interest from potential respondents, without
compromising the study. The interactive surveys were publicized by Chessgames.com
(based in Florida, USA) on their website but hosted on the author’s server. The basic
structure of the interactive combinations was created automatically using the ChessBase
9 database program ‘HTML+Javascript’ game export facility and edited using
Microsoft Frontpage. For the interested reader, details about these websites and
programs can easily be obtained through an Internet search (which would also be the
most up-to-date).
Surveys 1, 3 and 4 had the same instruction set. To avoid repetition, it is presented only
once in section 1.1. The instruction set for survey 2 differed slightly and is presented in
subsection 1.1.1. The actual combinations used for each survey were presented in a
separate frame to the left of the instruction set in the online version and could be played
through interactively (see pictures in Appendix F, section 1.6). Here, they are presented
in a set of tables in section 1.2 with the moves listed below each diagram. PGN
compatible versions of the combinations are provided in section 1.3. Section 1.4 shows
the control questions used in each survey.
283
Section 1.5 lists all the valid respondent data collected for each survey. Information
about gender (most were male), age (average around 30), and ratings (most were
unrated) did not prove useful. The author had hoped also to determine possible
differences in chess aesthetic evaluation between male and female players, but 99% of
respondents identified themselves as male, so data was insufficient. The author does not
however, suspect that any such differences would be statistically significant. Section 1.6
shows two screenshots of one of the actual online surveys so the reader has some idea
about what they really looked like because the surveys are no longer online.
1.1 Instruction Set (Surveys 1, 3 and 4)
Thank you for taking this survey. Its purpose is to assess human aesthetic perception in
chess to determine if there is a correlation with computational assessment. This research
could potentially enhance the versatility of chess database search engines, increase the
quality of automatic chess composition and improve game heuristics. Adaptation of the
technology to other chess variants and similar games is also foreseeable. The survey
should take about 20-30 minutes to complete.
Evaluate the 20 mate-in-3 combinations on the left and give a score between 1.0
(lowest) and 10.0 (highest) to each based on its BEAUTY. Your evaluation scores
must be precise to at least one decimal point (e.g. "5.7"). You can always fine-tune
them after reviewing all the combinations. Take as much as possible into account.
Press the "submit" button at the end of the page when you're done. Incomplete or
suspicious evaluations will not be processed and analyzed. If you would like to be
notified of the findings of this study, be sure to enter a valid e-mail address below.
284
Please Enter Your Aesthetic Scores in These Boxes
(all combinations must be attributed a score)
01: 02:
03: 04:
05: 06:
07: 08:
09: 10:
11: 12:
13: 14:
15: 16:
17: 18:
19: 20:
Enter Some Information about Yourself
(* indicates required fields)
Name: E-mail:
Age*: Gender*: male female
Chess Rating*:
Enter your most relevant chess rating, if any (e.g. "FIDE 1800", "USCF 1850"). Chess community ratings are also acceptable (e.g. "Chessgames.com 1875"). Otherwise enter "unrated".
285
Comments/Feedback about the Survey
(optional)
1.1.1 Instruction Set (Survey 2)
Thank you for taking this survey. Its purpose is to assess human aesthetic perception in
chess to determine if there is a correlation with computational assessment. This research
could potentially enhance the versatility of chess database search engines, increase the
quality of automatic chess composition and improve game heuristics. Adaptation of the
technology to other chess variants and similar games is also foreseeable. The survey
should take about 20-30 minutes to complete.
Evaluate each of the 10 pairs of mate-in-3 combinations on the left and choose the more
beautiful one. Then, enter a score between 1.0 (lowest) and 10.0 (highest) for each
based on its BEAUTY.
Press the "submit" button at the end of the page when you're done. Incomplete or
suspicious evaluations will not be processed and analyzed. If you would like to be
notified of the findings of this study, be sure to enter a valid e-mail address below.
286
Please Select the More Beautiful Combination in Each Pair
(all combinations must also be attributed a score)
01 02
03 04
05 06
07 08
09 10
11 12
13 14
15 16
17 18
19 20
287
Enter Some Information about Yourself
(* indicates required fields)
Name: E-mail:
Age*: Gender*: male female
Chess Rating*:
Enter your most relevant chess rating, if any (e.g. "FIDE 1800", "USCF 1850"). Chess community ratings are also acceptable (e.g. "Chessgames.com" 1875). Otherwise enter "unrated".
Comments/Feedback about the Survey
(optional)
288
1.2 The Combinations Rated
Survey 1 Combinations
XABCDEFGHY
8-+-+-+-+(
7+-+-+Q+-'
6p+q+-+pmk&
5+-+-+-+p%
4-+-+-+-+$
3+P+-+-+P#
2P+r+-+P+"
1+-+-tR-+K!
xabcdefghy
XABCDEFGHY
8-+r+-+-sN(
7zppsn-tRP+-'
6n+-mk-+p+&
5zp-+Pzp-vL-%
4-+-+-+-+$
3+RwQ-sN-+-#
2-mKP+PzPltr"
1+-+-+-+-!
xabcdefghy 1.Qf8+ Kg5 2.Re5+ Kh4 3.Qf4#
[1] 1.Nxg6 Re8 2.Rxb7 Nxd5 3.Rbd7#
[2] XABCDEFGHY
8Q+-+-+-+(
7+-+L+-+-'
6-+-zp-+-+&
5+pzpN+-+-%
4p+k+Pzp-+$
3+-+-zpl+-#
2-+P+-+-mK"
1+-+-+-+-!
xabcdefghy
XABCDEFGHY
8-+-sN-sn-+(
7vL-+p+-+q'
6P+-+-+-+&
5tRN+kzp-+-%
4K+-+pzp-+$
3+Pzp-sn-zpl#
2P+-+-+Q+"
1+-+-+L+-!
xabcdefghy 1.Qe8 Kd4 2.Qh8+ Kxe4 3.Nc3#
[3] 1.Kb4 Qe7+ 2.Nd6+ Kxd6 3.Bb8#
[4] XABCDEFGHY
8-+-+-+K+(
7+-+-+N+-'
6-+-+-+-+&
5+-+-+-+k%
4-+-+-+-+$
3+-+-+-wQ-#
2-+-+-zP-+"
1+-vl-tr-+-!
xabcdefghy
XABCDEFGHY
8r+-+l+r+(
7+-+-+-zp-'
6pmkpvLR+-zp&
5+-+-+-+-%
4-+-tR-+P+$
3+P+-+-+P#
2PmK-+-+-+"
1+-+-+-+-!
xabcdefghy 1.Kh7 Rh1 2.f3 Rg1 3.Qh2#
[5] 1.Rb4+ Ka5 2.Rb7 c5 3.Bc7#
[6]
289
XABCDEFGHY
8-+-+-+-+(
7+-zp-+-+-'
6-+L+-zp-+&
5zp-zp-+P+p%
4P+PzpP+-vL$
3+K+P+N+-#
2-+N+k+PtR"
1+-+-+-+-!
xabcdefghy
XABCDEFGHY
8-+-+-+-mk(
7zpp+-wQ-+p'
6-sn-+L+-wq&
5+-zpp+-+-%
4-+-+-+-zP$
3+-zP-+-+-#
2PzP-+-+PmK"
1+-+-+-+-!
xabcdefghy 1.Kb2 Kxd3 2.g3 Kxc4 3.Bb5#
[7] 1.Qe8+ Kg7 2.Qf7+ Kh8 3.Qg8#
[8] XABCDEFGHY
8-+-+k+-tr(
7zp-+n+p+p'
6-+Q+-zp-+&
5+-+-+-+-%
4-tr-zP-+-sN$
3+-zpLzP-+-#
2Pwq-+-zPPzP"
1+-+lmK-+R!
xabcdefghy
XABCDEFGHY
8-+-+-+-+(
7+-+-+-+-'
6P+-+-+-+&
5+P+-+-sN-%
4K+-zP-mkPzP$
3+-sN-+-+Q#
2P+-zP-+-+"
1+-+-tr-+l!
xabcdefghy 1.Qc8+ Ke7 2.Nf5+ Ke6 3.Qc6#
[9] 1.a3 Re5 2.Nge4 Rxe4 3.Nd5#
[10] XABCDEFGHY
8-+Q+-+-+(
7+-+-+-+-'
6-+-+-+p+&
5zP-+-+-zp-%
4-+-+-+Pzp$
3+-tR-+-+P#
2-wQK+-+k+"
1tr-+-+-+-!
xabcdefghy
XABCDEFGHY
8-+-+-+-+(
7sNr+pzp-sN-'
6lzp-+-+-+&
5vl-+PtR-+-%
4ptrkvL-zP-+$
3+n+R+-+-#
2L+-+K+-+"
1+-+-+-+-!
xabcdefghy 1.Qxa1 Kf2 2.Qc5+ Kg2 3.Qag1#
[11] 1.Ba1 Kc5 2.d6+ Kc4 3.Rd4#
[12] XABCDEFGHY
8-+-+-+R+(
7+p+-+p+-'
6r+-zp-mk-+&
5zp-+Pzpq+-%
4-+-+-vl-zp$
3+-zP-+-+-#
2PzP-+-+-zP"
1mK-+-+RwQ-!
xabcdefghy
XABCDEFGHY
8-+R+-+-+(
7+-+-+-+-'
6-+-mk-+-+&
5+-+L+-+K%
4-+-vLp+-+$
3+-+p+-+-#
2Q+-+-zp-vl"
1+r+-+-+-!
xabcdefghy 1.Qg7+ Ke7 2.Qf8+ Kf6 3.Qd8#
[13] 1.Qxf2 Kxd5 2.Qf5+ Kxd4 3.Qc5#
[14]
290
XABCDEFGHY
8-+-+ksnQ+(
7+-+-+-+-'
6-+-mK-+-+&
5+-+-+-+-%
4-+-+-+-+$
3+-+-+-+-#
2-+-+-+-+"
1+-+-+-+-!
xabcdefghy
XABCDEFGHY
8-+-snr+k+(
7zppzpqzp-+-'
6-+-+Nvl-wQ&
5+-+-+-+-%
4-+-zPR+-zp$
3+-zP-+-+P#
2PzP-+-zPP+"
1tR-+-+-mK-!
xabcdefghy 1.Qg7 Ng6 2.Qf6 Nf8 3.Qe7#
[15] 1.Qg6+ Kh8 2.Rxh4+ Bxh4 3.Qg7#
[16] XABCDEFGHY
8-+-+-+-+(
7+-+-+p+-'
6-+pmK-zP-sn&
5+-+-tR-+R%
4-zpPzp-mk-+$
3+-+-sN-+P#
2-+-+-+P+"
1+-+-vlN+n!
xabcdefghy
XABCDEFGHY
8-+-+n+r+(
7+ptr-+-+-'
6p+-zpQ+-zp&
5+-+-zPq+k%
4-zP-+-+-+$
3+-+-+-zPP#
2P+-+-+-mK"
1+-+-+-+-!
xabcdefghy 1.Nc2 Bd2 2.Rh4+ Ng4 3.Rxg4#
[17] 1.Qxf5+ Rg5 2.g4+ Kh4 3.Qf2#
[18] XABCDEFGHY
8-+-tr-mkr+(
7zpp+n+-+-'
6-wq-+p+Q+&
5+-+pzP-zp-%
4-+-zP-+-+$
3+-+-+-zP-#
2PzP-+-+-+"
1+-tR-+-mK-!
xabcdefghy
XABCDEFGHY
8-+-+-wQ-+(
7+p+-+R+-'
6-+k+-+p+&
5+-+-+-zP-%
4-zpP+-+-+$
3+Psn-+-+P#
2-+-+r+-+"
1+-+-+-+K!
xabcdefghy 1.Rf1+ Nf6 2.Rxf6+ Ke7 3.Qf7#
[19] 1.Qc8+ Kd6 2.Qc7+ Ke6 3.Qe7#
[20]
291
Survey 2 Combinations
XABCDEFGHY
8r+-wqr+k+(
7zplzpnvlNzpp'
6-zp-+Qsn-+&
5+-+-+-+-%
4-+-zP-+-+$
3+-+L+-+-#
2PzPP+-zPPzP"
1tR-vL-tR-mK-!
xabcdefghy
XABCDEFGHY
8-+-+-+r+(
7zp-zp-+-+k'
6-zpPzp-+r+&
5+-+P+p+q%
4-+-vL-+-+$
3+-+-+-+Q#
2PzP-+-sN-zP"
1+-+-+-+K!
xabcdefghy 1.Nh6+ Kh8 2.Qg8+ Nxg8 3.Nf7#
[1] 1.Qxh5+ Rh6 2.Qf7+ Rg7 3.Qxg7#
[2] XABCDEFGHY
8r+-+kvl-tr(
7zppzp-zpN+p'
6-wq-+-sn-+&
5+-+P+L+Q%
4-+P+-+-+$
3+-+-+-+-#
2PzP-+-zP-zP"
1sn-vLK+-+R!
xabcdefghy
XABCDEFGHY
8-+-+-wQ-+(
7zp-+-zp-+p'
6-zpq+n+p+&
5zP-+pzP-mk-%
4-zP-zP-+-+$
3+-+-+-+-#
2-+-+L+PzP"
1+-+-+-mK-!
xabcdefghy 1.Nd6+ Kd8 2.Qe8+ Nxe8 3.Nf7#
[3] 1.h4+ Kxh4 2.Qh6+ Kg3 3.Qh2#
[4] XABCDEFGHY
8r+-+-trk+(
7+q+-+p+-'
6-+-vlpwQ-+&
5zp-+-+-+-%
4-zp-zP-zP-+$
3+-zP-sNl+-#
2PzPL+N+-zP"
1tR-+-+K+-!
xabcdefghy
XABCDEFGHY
8r+r+-+-+(
7+l+-wqp+p'
6-+-sNpmk-+&
5zp-sn-+-zp-%
4-zp-+-+-wQ$
3+-+-zP-+-#
2-+-+-zPPzP"
1+-+RmKL+R!
xabcdefghy 1.Qg5+ Kh8 2.Qh6+ Kg8 3.Qh7#
[5] 1.Qh6+ Ke5 2.f4+ gxf4 3.exf4#
[6]
292
XABCDEFGHY
8r+-+-+k+(
7zpp+q+p+p'
6-+nzp-vL-+&
5+-zp-sn-+Q%
4-+P+-+-+$
3+-+-+-+-#
2PzP-+-+PzP"
1tR-+-+RmK-!
xabcdefghy
XABCDEFGHY
8-+R+-+-+(
7tr-+-+-+k'
6-+-+LmK-+&
5+-+-+-+-%
4-+-+-+-zp$
3+-+-+-+P#
2-+-+-+-+"
1+-+-+-+-!
xabcdefghy 1.Qg5+ Ng6 2.Qh6 a5 3.Qg7#
[7] 1.Bf5+ Kh6 2.Rh8+ Rh7 3.Rxh7#
[8] XABCDEFGHY
8r+l+-mk-+(
7zpp+-+-zpp'
6-+-+-+-+&
5+-zpn+-+-%
4-+L+pvLq+$
3+-zP-+-+-#
2PzP-+-+P+"
1tR-+-+RmK-!
xabcdefghy
XABCDEFGHY
8-+-tr-+-+(
7+-+-+k+p'
6-+-+psNpwQ&
5+-+p+p+-%
4-+-zP-+-+$
3+-+-+-vLP#
2q+-+-zPPmK"
1+-+-+-+-!
xabcdefghy 1.Bd6+ Ke8 2.Bb5+ Bd7 3.Rf8#
[9] 1.Qxh7+ Kxf6 2.Be5+ Kg5 3.f4#
[10] XABCDEFGHY
8-+-+-+-+(
7+-+-+-+-'
6-+-+Kzp-+&
5tR-+-+-+-%
4-+-+-+-+$
3+-+-wQ-+-#
2-+-+-+r+"
1+-+-+-+k!
xabcdefghy
XABCDEFGHY
8-+-+-vL-+(
7+K+-tR-+l'
6-+-mk-+-+&
5+P+-+-tR-%
4-+-trp+-+$
3+-sN-+-+-#
2-+n+-vl-sn"
1+-+-+-+-!
xabcdefghy 1.Qe4 Kh2 2.Qh4+ Kg1 3.Ra1#
[11] 1.Nd5 Kc5 2.Rc7+ Kxb5 3.Nc3#
[12] XABCDEFGHY
8L+-+-+-+(
7+q+NwQ-mK-'
6-+n+-+-+&
5+-+nzp-+-%
4-vlp+kzP-sN$
3+-zP-+-+-#
2-+-zPP+-+"
1+-+-+R+-!
xabcdefghy
XABCDEFGHY
8q+-+-vL-+(
7+-+-zP-sn-'
6-+-+k+-+&
5+l+-sNrzP-%
4-wQ-+-zP-+$
3+-+-mK-+-#
2-+-+-+-+"
1+-+-+-+-!
xabcdefghy 1.Rf3 Nxc3 2.Nf6+ Kd4 3.Nf5#
[13] 1.Qd6+ Kxd6 2.e8N+ Kd5 3.Nc7#
[14]
293
XABCDEFGHY
8-+-+-+-+(
7vl-+-+-+-'
6n+N+-mK-+&
5+-+-+-+-%
4r+P+k+-+$
3trp+-+R+-#
2l+-tR-+-+"
1+-sn-+-+L!
xabcdefghy
XABCDEFGHY
8r+-+-snk+(
7tr-zp-+N+q'
6-+-+-zP-+&
5vl-+-+-+-%
4-+-+-+-+$
3+Q+-+-+-#
2-+-+-+-+"
1mK-+-+-+R!
xabcdefghy 1.Rd5 Ra5 2.c5 Kxd5 3.Rc3#
[15] 1.Nh6+ Kh8 2.Qg8+ Qxg8 3.Nf7#
[16] XABCDEFGHY
8-+-+-+-+(
7+-zp-+-+-'
6-+p+l+-+&
5mK-mk-vL-+-%
4-+-+-+-+$
3+L+-+-+-#
2-zP-tR-+-+"
1+-+-+-+-!
xabcdefghy
XABCDEFGHY
8-+-+-+-+(
7+-+p+-+-'
6R+-sN-+r+&
5mK-+-+-vL-%
4-+-+p+-+$
3mkp+-+p+-#
2-tr-snl+-+"
1+nsN-vlq+-!
xabcdefghy 1.Rd7 Bxb3 2.Rd4 Ba2 3.b4#
[17] 1.Be7 Nc4+ 2.Kb5+ Na5+ 3.Nc4#
[18] XABCDEFGHY
8-+-+-vL-+(
7+-+-+-zP-'
6-zp-+-+-+&
5+-+-+-+-%
4-sN-+-+-+$
3mkP+-+-+-#
2-+K+-+-+"
1+-+-+-+-!
xabcdefghy
XABCDEFGHY
8-+-+-+-+(
7+-zp-+N+-'
6-+-zp-+-+&
5zp-+-+-wQ-%
4KsnPmkL+rtR$
3+-+-+-+-#
2-+PvL-vl-sn"
1+-+-+-+-!
xabcdefghy 1.g8N b5 2.Ne7 Kxb4 3.Nc6#
[19] 1.Kb5 Rxg5+ 2.Bf5+ Bxh4 3.c3#
[20]
294
Survey 3 Combinations
XABCDEFGHY
8r+-tr-vl-+(
7zP-+-+-zpp'
6-+-zPkzp-+&
5+-snR+-+-%
4-+L+-+-zP$
3+-+-+-+-#
2-+-+-+P+"
1+-+R+-+K!
xabcdefghy
XABCDEFGHY
8-+-+-+k+(
7+-+-+p+p'
6p+r+-+-+&
5sn-+-+P+-%
4P+-zp-+-wQ$
3+-+P+-zP-#
2q+-+-+-zP"
1+-+-tR-mK-!
xabcdefghy 1.Rxc5+ Kd7 2.Rc7+ Ke8 3.Bf7#
[1] 1.Re8+ Kg7 2.Qg5+ Rg6 3.f6#
[2] XABCDEFGHY
8-+-trk+-+(
7tR-+-+-tRp'
6-+-+-+p+&
5+-+l+pzP-%
4-vL-+-zP-+$
3+n+-+-+-#
2-zP-+-+-zP"
1+-+-+-mK-!
xabcdefghy
XABCDEFGHY
8-+-wQ-vlk+(
7+-+-vLp+p'
6-zp-+-+p+&
5+P+-+-+-%
4-+-zp-+-+$
3+-+P+-zPL#
2-+-+-+-wq"
1+R+-+R+K!
xabcdefghy 1.Rae7+ Kf8 2.Ref7+ Ke8 3.Rf8#
[3] 1.Kxh2 h5 2.Qxf8+ Kh7 3.Rxf7#
[4] XABCDEFGHY
8r+l+r+k+(
7zp-wq-+p+p'
6-zp-+nwQ-+&
5+-+-+N+-%
4-+-tRp+-+$
3+-+-+-zPL#
2P+-+-zP-zP"
1+-+-+-mK-!
xabcdefghy
XABCDEFGHY
8-tr-wq-mk-+(
7+-+-+-+-'
6l+n+rvl-sN&
5+-zpp+-+P%
4pzp-+-+Q+$
3+-+P+NzP-#
2PzPP+-zP-+"
1tR-+-+-mK-!
xabcdefghy 1.Nh6+ Kf8 2.Qh8+ Ke7 3.Nf5#
[5] 1.Qg8+ Ke7 2.Qf7+ Kd6 3.Nf5#
[6]
295
XABCDEFGHY
8r+q+-+-tr(
7zpp+-+n+p'
6-+p+lsN-mk&
5+-zPp+-zp-%
4-zP-+-zPP+$
3+-+R+-+-#
2PwQ-+L+-zP"
1+-+-+RmK-!
xabcdefghy
XABCDEFGHY
8r+-tr-mknwQ(
7zpl+-+-zpL'
6-zp-+pzp-+&
5+-+-tR-vL-%
4-+-wq-+-+$
3+-+-+-+-#
2P+-+-zPPzP"
1+-+-tR-mK-!
xabcdefghy 1.Ng8+ Qxg8 2.Qf6+ Qg6 3.Rh3#
[7] 1.Qxg8+ Ke7 2.Qxe6+ Kf8 3.Qg8#
[8] XABCDEFGHY
8r+-+-snk+(
7+-+ltrp+-'
6-+-zp-+p+&
5zp-zpPzp-wq-%
4-+L+P+-tR$
3+-+-+-+-#
2PzP-+-zP-wQ"
1+K+-+-+R!
xabcdefghy
XABCDEFGHY
8-+-+-+-+(
7+-+-+-+-'
6-+-+-+R+&
5zpptrkzpN+-%
4-+-+-+-zp$
3+-zP-+-+P#
2P+-+-mKP+"
1+-+-+-+-!
xabcdefghy 1.Rh8+ Kg7 2.Rg8+ Kf6 3.Qh8#
[9] 1.Ne3+ Ke4 2.Ke2 Rxc3 3.Rg4#
[10] XABCDEFGHY
8r+-wqrvlk+(
7+-+-+-zp-'
6-+lzPR+-zp&
5+-zp-+L+Q%
4PsnN+-+-+$
3+-vL-+-zP-#
2-+-+-+-zP"
1+-+-tR-mK-!
xabcdefghy
XABCDEFGHY
8-+-+-+-+(
7+-+-+kvlp'
6-+-+-zpp+&
5+-vLr+-+-%
4-+Q+-+-+$
3+-+-+-zP-#
2-+-+-zP-zP"
1wq-+-sN-mK-!
xabcdefghy 1.Qg6 Rxe6 2.Bxe6+ Kh8 3.Qxh6#
[11] 1.Qxd5+ Ke8 2.Qe6+ Kd8 3.Bb6#
[12] XABCDEFGHY
8-+-wQ-+-+(
7+-mK-tR-+-'
6-+-+-mk-+&
5+-zP-+-+-%
4-+-+-+-+$
3+-+-+-+-#
2-+-+-+-+"
1+-+-+-+-!
xabcdefghy
XABCDEFGHY
8-+-+R+-+(
7+-+-tR-+p'
6-zp-+-+p+&
5+P+p+-mk-%
4-+-vl-+P+$
3+-+-+-mK-#
2-+-trLtr-zP"
1+-+-+-+-!
xabcdefghy 1.Qf8+ Kg6 2.Rg7+ Kh6 3.Qh8#
[13] 1.h4+ Kf6 2.g5+ Kf5 3.Bg4#
[14]
296
XABCDEFGHY
8-+-+ntR-+(
7zp-+-+R+p'
6-+q+-+pmk&
5+-+-zp-+-%
4-+-+N+P+$
3+-+P+-+-#
2-+-+-+-zP"
1+-+-+-+K!
xabcdefghy
XABCDEFGHY
8-+-+-+-+(
7+-+-+-+-'
6-+-+-+r+&
5+-+-wQ-+-%
4-+-+KzPk+$
3zp-+-+-+p#
2-+-+-+-zP"
1+-+-+-+-!
xabcdefghy 1.g5+ Kh5 2.Rxh7+ Kg4 3.h3#
[15] 1.Qf5+ Kh4 2.Qxg6 a2 3.Qg5#
[16] XABCDEFGHY
8r+-+-tr-+(
7+p+R+-+-'
6-+p+-mk-+&
5+-+-+-sNP%
4-zPP+-zPP+$
3+-vl-sN-+-#
2-+-+P+-+"
1+-+-+-mK-!
xabcdefghy
XABCDEFGHY
8rsn-mkl+R+(
7zpp+-wq-+-'
6-+-zp-+-vL&
5+NzpPzpLwQ-%
4-+P+-+-+$
3+-+-+-+P#
2PzP-+-+-mK"
1+-+-+-+-!
xabcdefghy 1.Ne4+ Ke6 2.Nc5+ Kf6 3.g5#
[17] 1.Qxe7+ Kxe7 2.Bg5+ Kf7 3.Be6#
[18] XABCDEFGHY
8-+Q+lsnk+(
7+-+-+-zpp'
6-+-+pzP-+&
5+-vLp+-+-%
4-zp-+-zP-+$
3+-+-+-+-#
2-zP-+-+-zP"
1+-+-+-mK-!
xabcdefghy
XABCDEFGHY
8-+r+r+-mk(
7tR-+-+R+-'
6-zp-+-+-vL&
5+-+N+-+-%
4P+-+n+-+$
3+-+-+-+-#
2-+p+-+PzP"
1+-+-+-mK-!
xabcdefghy 1.Qxe8 h5 2.Qxf8+ Kh7 3.Qxg7#
[19] 1.Rh7+ Kg8 2.Rag7+ Kf8 3.Rg6#
[20]
297
Survey 4 Combinations
XABCDEFGHY
8-+-mK-+-+(
7+-+N+-+-'
6-+-zp-+-+&
5+-+k+-+-%
4-+-vL-+-+$
3+-+-+-+-#
2-+-+-+-+"
1+-+-+-wQ-!
xabcdefghy
XABCDEFGHY
8l+r+kvl-+(
7zprzp-zppzp-'
6-zpp+N+-+&
5+-+-+-+-%
4-+-+-+-+$
3+-+-wQ-+-#
2-+-tR-+-+"
1+n+-+-+K!
xabcdefghy 1.Qd1 Kc6 2.Qb3 d5 3.Qb6#
[1] 1.Qh3 fxe6 2.Qh5+ g6 3.Qxg6#
[2] XABCDEFGHY
8Q+-+-tR-vl(
7+r+-zpn+-'
6-+NzpNzPltR&
5+-+P+k+-%
4LzP-+-+-+$
3+-+PmK-+P#
2p+-+-+-+"
1+-+-+-+-!
xabcdefghy
XABCDEFGHY
8-+R+-+n+(
7+-+-+L+-'
6-+-+-zp-zp&
5+-+-+p+-%
4-zp-zPN+-+$
3+-+N+-+-#
2-+-+k+-mK"
1+lwQ-+-+n!
xabcdefghy 1.Bc2 Kxf6 2.Rxf7+ Kxf7 3.Qf8#
[3] 1.Rd8 Kxd3 2.Qd2+ Kxe4 3.Bd5#
[4] XABCDEFGHY
8-+-+-+-+(
7+-+-+-+-'
6K+-snn+-+&
5+-mk-+-+-%
4psN-+-+-+$
3zP-+P+N+-#
2-vL-+-+-+"
1+-+-+-+-!
xabcdefghy
XABCDEFGHY
8-+l+-+-+(
7+p+p+-+-'
6-zP-mK-zp-+&
5zp-zp-zpQ+-%
4k+-+N+-+$
3+-+-+-+-#
2-+P+-+-+"
1+-vL-+-+-!
xabcdefghy 1.Bd4+ Nxd4 2.Ng5 Nc8 3.Ne4#
[5] 1.Qe6 Kb5 2.Qb3+ Ka6 3.Nxc5#
[6]
298
XABCDEFGHY
8-+-vl-+-+(
7+-+-zpK+-'
6p+-trpzpN+&
5tr-+k+Pzp-%
4L+Rzpp+-+$
3+P+-+-+-#
2p+-sNP+-+"
1+-+-wQ-+-!
xabcdefghy
XABCDEFGHY
8q+-+-+-mk(
7+-+-+-zp-'
6-+-+N+P+&
5+-+-+-+-%
4-+-+-+-+$
3vL-+-+-+-#
2p+-+P+-+"
1mK-+Q+-+-!
xabcdefghy 1.Nb1 Rc6 2.Qb4 Rxc4 3.bxc4#
[7] 1.Qf1 Kg8 2.Qf7+ Kh8 3.Qxg7#
[8] XABCDEFGHY
8-+K+-+-+(
7+-+-+-+-'
6P+-+LvL-+&
5+-+-zP-+-%
4-+-zppmkN+$
3+-zP-+r+-#
2-+-+-wQRzP"
1+-vl-+-+-!
xabcdefghy
XABCDEFGHY
8-+-+-+-+(
7+-+-+-+L'
6-+-zp-zp-+&
5vL-+-mk-+-%
4-sN-zp-wqQ+$
3sn-+-zpPsn-#
2-zP-sN-+K+"
1+-+-+-+-!
xabcdefghy 1.Bd8 dxc3 2.Qa7 e3 3.Qd4#
[9] 1.Qg6 Qxf3+ 2.Nxf3+ Ke6 3.Qe8#
[10] XABCDEFGHY
8-wQ-+-+-sN(
7+-+-+-+-'
6-+k+-+-+&
5+-+-+-+-%
4-+-+-+-+$
3+N+-+-+-#
2-+-+LmKl+"
1+-+-+-+-!
xabcdefghy
XABCDEFGHY
8-vL-+-+-+(
7+-+-vlKzp-'
6psnQ+-+L+&
5zP-+N+-+-%
4-tRPmkrzPRtr$
3+-zp-zp-+-#
2q+p+-zP-+"
1+-sN-+l+-!
xabcdefghy 1.Ng6 Kd7 2.Bg4+ Kc6 3.Ne7#
[11] 1.Nc7 Rxf4+ 2.Qf6+ Kc5 3.Nxa6#
[12] XABCDEFGHY
8-+-+-+-+(
7+-+-+-+-'
6-+-+-+-+&
5+-+-+-+-%
4-+-wQ-+-+$
3+-zpp+-mK-#
2n+-+k+-+"
1sN-+-+-+N!
xabcdefghy
XABCDEFGHY
8-trr+nwQl+(
7+n+-+-+-'
6pzP-zPp+-+&
5+pzPNvl-+-%
4-zpk+N+LtR$
3+p+-zPP+-#
2-zP-+-+-+"
1+-+R+-+K!
xabcdefghy 1.Kf4 Kd1 2.Qxd3+ Kc1 3.Qc2#
[13] 1.d7 Rxc5 2.Bf5 Rxd5 3.Nd2#
[14]
299
XABCDEFGHY
8K+-+-wQ-+(
7zp-+LzP-zp-'
6-+-tRp+-vl&
5+-+pmk-zPP%
4-+-tRN+-+$
3+P+-zp-+l#
2-zP-+P+-+"
1+-+-+-+-!
xabcdefghy
XABCDEFGHY
8-+-+-+-+(
7+-+p+-+-'
6-+lvL-+-+&
5zP-wQp+-+-%
4k+-+-+-+$
3+r+L+-+-#
2-+-+nmK-+"
1sn-+-+-+-!
xabcdefghy 1.Ra6 Kxd4 2.Qf4 dxe4 3.Qd6#
[15] 1.Ke1 Rb1+ 2.Bxb1 Nc2+ 3.Bxc2#
[16] XABCDEFGHY
8-+-+l+-+(
7+-+-mK-+p'
6-+-+-+ptr&
5zpP+kzp-+-%
4-+-+-+-zp$
3sn-zpLwQR+-#
2n+-+-zP-+"
1+-+-+-+-!
xabcdefghy
XABCDEFGHY
8-+-+-+-+(
7+-+-+N+-'
6-+k+-+-+&
5+-+-zp-+-%
4-vL-+-+-+$
3mK-+Q+-+-#
2-+-+-+-+"
1+-+-+-+-!
xabcdefghy 1.Rf4 exf4 2.Qe6+ Kc5 3.Qd6#
[17] 1.Qh7 Kb7 2.Nd6+ Ka8 3.Qb7#
[18] XABCDEFGHY
8-+-+-+-+(
7+-+-+-+-'
6-sNn+-+-+&
5+p+-+-+p%
4-+-mk-zpP+$
3wQ-+-+-+-#
2-zp-+-+-+"
1+K+-tR-+-!
xabcdefghy
XABCDEFGHY
8-+L+-+-+(
7+p+-sN-+-'
6-mK-zp-+-+&
5+-+pzp-+-%
4-+-+k+-+$
3+-+-+p+-#
2-+R+-zP-+"
1vL-+-sN-+-!
xabcdefghy 1.Nd7 Kc4 2.Re4+ Kd5 3.Nf6#
[19] 1.Bd4 Kxd4 2.Bxb7 Ke4 3.Rc4#
[20]
300
1.3 The Combinations Rated (PGN Compatible)
The following is the PGN compatible equivalent of the combinations shown in the
previous section. The combinations are not presented in the typical PGN format which
includes additional information about the game (e.g. player names, tournament, date
etc.), as this was considered superfluous for the purpose of this section. The ones here
are presented in sequence from positions 1 through 20 for each survey set but not
numbered as such or in table form to preserve PGN compatibility. Researchers
interested in using these (human player) aesthetically-rated combinations may simply
scan the page using optical character recognition and save the contents (between the
dotted lines, ---, for each survey set) in a text file with a *.pgn extension.
It can then be opened automatically using software that supports the format. The author
tested it using the program, ChessBase 10 (details about it are best obtained through a
simple and up-to-date Internet search). This is probably easier than setting up each
position manually and playing out the moves in another program. Given that reliable
data of this nature are scarce or perhaps even unprecedented, and that such surveys are
not easy to carry out, the author hopes these combinations and their individual
respondent ratings (see Appendix F, section 1.5), or average respondent ratings (see
chapter 6, Tables 6.10, 6.11, 6.13 and 6.14 for surveys 1 through 4, respectively), will
prove useful in other research endeavours as well. Chances are, the game of chess as it
is today, will be around for many centuries to come, so these combinations and their
associated human chess-player aesthetic ratings will be perfectly viable for a long time.
301
Survey 1 Combinations (PGN)
[FEN "8/5Q2/p1q3pk/7p/8/1P5P/P1r3P1/4R2K w - - 0 1"] 1. Qf8+ Kg5 2. Re5+ Kh4 3. Qf4# 1-0 [FEN "2r4N/ppn1RP2/n2k2p1/p2Pp1B1/8/1RQ1N3/1KP1PPbr/8 w - - 0 1"] 1. Nxg6 Re8 2. Rxb7 Nxd5 3. Rbd7# 1-0 [FEN "Q7/3B4/3p4/1ppN4/p1k1Pp2/4pb2/2P4K/8 w - - 0 1"] 1. Qe8 Kd4 2. Qh8+ Kxe4 3. Nc3# 1-0 [FEN "3N1n2/B2p3q/P7/RN1kp3/K3pp2/1Pp1n1pb/P5Q1/5B2 w - - 0 1"] 1. Kb4 Qe7+ 2. Nd6+ Kxd6 3. Bb8# 1-0 [FEN "6K1/5N2/8/7k/8/6Q1/5P2/2b1r3 w - - 0 1"] 1. Kh7 Rh1 2. f3 Rg1 3. Qh2# 1-0 [FEN "r3b1r1/6p1/pkpBR2p/8/3R2P1/1P5P/PK6/8 w - - 0 1"] 1. Rb4+ Ka5 2. Rb7 c5 3. Bc7# 1-0 [FEN "8/2p5/2B2p2/p1p2P1p/P1PpP2B/1K1P1N2/2N1k1PR/8 w - - 0 1"] 1. Kb2 Kxd3 2. g3 Kxc4 3. Bb5# 1-0 [FEN "7k/pp2Q2p/1n2B2q/2pp4/7P/2P5/PP4PK/8 w - - 0 1"] 1. Qe8+ Kg7 2. Qf7+ Kh8 3. Qg8# 1-0 [FEN "4k2r/p2n1p1p/2Q2p2/8/1r1P3N/2pBP3/Pq3PPP/3bK2R w Kk - 0 1"] 1. Qc8+ Ke7 2. Nf5+ Ke6 3. Qc6# 1-0 [FEN "8/8/P7/1P4N1/K2P1kPP/2N4Q/P2P4/4r2b w - - 0 1"] 1. a3 Re5 2. Nge4 Rxe4 3. Nd5# 1-0 [FEN "2Q5/8/6p1/P5p1/6Pp/2R4P/1QK3k1/r7 w - - 0 1"] 1. Qxa1 Kf2 2. Qc5+ Kg2 3. Qag1# 1-0 [FEN "8/Nr1pp1N1/bp6/b2PR3/prkB1P2/1n1R4/B3K3/8 w - - 0 1"] 1. Ba1 Kc5 2. d6+ Kc4 3. Rd4# 1-0 [FEN "6R1/1p3p2/r2p1k2/p2Ppq2/5b1p/2P5/PP5P/K4RQ1 w - - 0 1"] 1. Qg7+ Ke7 2. Qf8+ Kf6 3. Qd8# 1-0 [FEN "2R5/8/3k4/3B3K/3Bp3/3p4/Q4p1b/1r6 w - - 0 1"] 1. Qxf2 Kxd5 2. Qf5+ Kxd4 3. Qc5# 1-0 [FEN "4knQ1/8/3K4/8/8/8/8/8 w - - 0 1"] 1. Qg7 Ng6 2. Qf6 Nf8 3. Qe7# 1-0 [FEN "3nr1k1/pppqp3/4Nb1Q/8/3PR2p/2P4P/PP3PP1/R5K1 w - - 0 1"] 1. Qg6+ Kh8 2. Rxh4+ Bxh4 3. Qg7# 1-0 [FEN "8/5p2/2pK1P1n/4R2R/1pPp1k2/4N2P/6P1/4bN1n w - - 0 1"] 1. Nc2 Bd2 2. Rh4+ Ng4 3. Rxg4# 1-0 [FEN "4n1r1/1pr5/p2pQ2p/4Pq1k/1P6/6PP/P6K/8 w - - 0 1"] 1. Qxf5+ Rg5 2. g4+ Kh4 3. Qf2# 1-0 [FEN "3r1kr1/pp1n4/1q2p1Q1/3pP1p1/3P4/6P1/PP6/2R3K1 w - - 0 1"] 1. Rf1+ Nf6 2. Rxf6+ Ke7 3. Qf7# 1-0 [FEN "5Q2/1p3R2/2k3p1/6P1/1pP5/1Pn4P/4r3/7K w - - 0 1"] 1. Qc8+ Kd6 2. Qc7+ Ke6 3. Qe7# 1-0
302
Survey 2 Combinations (PGN)
[FEN "r2qr1k1/pbpnbNpp/1p2Qn2/8/3P4/3B4/PPP2PPP/R1B1R1K1 w - - 0 1"] 1. Nh6+ Kh8 2. Qg8+ Nxg8 3. Nf7# 1-0 [FEN "6r1/p1p4k/1pPp2r1/3P1p1q/3B4/7Q/PP3N1P/7K w - - 0 1"] 1. Qxh5+ Rh6 2. Qf7+ Rg7 3. Qxg7# 1-0 [FEN "r3kb1r/ppp1pN1p/1q3n2/3P1B1Q/2P5/8/PP3P1P/n1BK3R w - - 0 1"] 1. Nd6+ Kd8 2. Qe8+ Nxe8 3. Nf7# 1-0 [FEN "5Q2/p3p2p/1pq1n1p1/P2pP1k1/1P1P4/8/4B1PP/6K1 w - - 0 1"] 1. h4+ Kxh4 2. Qh6+ Kg3 3. Qh2# 1-0 [FEN "r4rk1/1q3p2/3bpQ2/p7/1p1P1P2/2P1Nb2/PPB1N2P/R4K2 w - - 0 1"] 1. Qg5+ Kh8 2. Qh6+ Kg8 3. Qh7# 1-0 [FEN "r1r5/1b2qp1p/3Npk2/p1n3p1/1p5Q/4P3/5PPP/3RKB1R w K - 0 1"] 1. Qh6+ Ke5 2. f4+ gxf4 3. exf4# 1-0 [FEN "r5k1/pp1q1p1p/2np1B2/2p1n2Q/2P5/8/PP4PP/R4RK1 w - - 0 1"] 1. Qg5+ Ng6 2. Qh6 a5 3. Qg7# 1-0 [FEN "2R5/r6k/4BK2/8/7p/7P/8/8 w - - 0 1"] 1. Bf5+ Kh6 2. Rh8+ Rh7 3. Rxh7# 1-0 [FEN "r1b2k2/pp4pp/8/2pn4/2B1pBq1/2P5/PP4P1/R4RK1 w - - 0 1"] 1. Bd6+ Ke8 2. Bb5+ Bd7 3. Rf8# 1-0 [FEN "3r4/5k1p/4pNpQ/3p1p2/3P4/6BP/q4PPK/8 w - - 0 1"] 1. Qxh7+ Kxf6 2. Be5+ Kg5 3. f4# 1-0 [FEN "8/8/4Kp2/R7/8/4Q3/6r1/7k w - - 0 1"] 1. Qe4 Kh2 2. Qh4+ Kg1 3. Ra1# 1-0 [FEN "5B2/1K2R2b/3k4/1P4R1/3rp3/2N5/2n2b1n/8 w - - 0 1"] 1. Nd5 Kc5 2. Rc7+ Kxb5 3. Nc3# 1-0 [FEN "B7/1q1NQ1K1/2n5/3np3/1bp1kP1N/2P5/3PP3/5R2 w - - 0 1"] 1. Rf3 Nxc3 2. Nf6+ Kd4 3. Nf5# 1-0 [FEN "q4B2/4P1n1/4k3/1b2NrP1/1Q3P2/4K3/8/8 w - - 0 1"] 1. Qd6+ Kxd6 2. e8=N+ Kd5 3. Nc7# 1-0 [FEN "8/b7/n1N2K2/8/r1P1k3/rp3R2/b2R4/2n4B w - - 0 1"] 1. Rd5 Ra5 2. c5 Kxd5 3. Rc3# 1-0 [FEN "r4nk1/r1p2N1q/5P2/b7/8/1Q6/8/K6R w - - 0 1"] 1. Nh6+ Kh8 2. Qg8+ Qxg8 3. Nf7# 1-0 [FEN "8/2p5/2p1b3/K1k1B3/8/1B6/1P1R4/8 w - - 0 1"] 1. Rd7 Bxb3 2. Rd4 Ba2 3. b4# 1-0 [FEN "8/3p4/R2N2r1/K5B1/4p3/kp3p2/1r1nb3/1nN1bq2 w - - 0 1"] 1. Be7 Nc4+ 2. Kb5+ Na5+ 3. Nc4# 1-0 [FEN "5B2/6P1/1p6/8/1N6/kP6/2K5/8 w - - 0 1"] 1. g8=N b5 2. Ne7 Kxb4 3. Nc6# 1-0 [FEN "8/2p2N2/3p4/p5Q1/KnPkB1rR/8/2PB1b1n/8 w - - 0 1"] 1. Kb5 Rxg5+ 2. Bf5+ Bxh4 3. c3# 1-0
303
Survey 3 Combinations (PGN)
[FEN "r2r1b2/P5pp/3Pkp2/2nR4/2B4P/8/6P1/3R3K w - - 0 1"] 1. Rxc5+ Kd7 2. Rc7+ Ke8 3. Bf7# 1-0 [FEN "6k1/5p1p/p1r5/n4P2/P2p3Q/3P2P1/q6P/4R1K1 w - - 0 1"] 1. Re8+ Kg7 2. Qg5+ Rg6 3. f6# 1-0 [FEN "3rk3/R5Rp/6p1/3b1pP1/1B3P2/1n6/1P5P/6K1 w - - 0 1"] 1. Rae7+ Kf8 2. Ref7+ Ke8 3. Rf8# 1-0 [FEN "3Q1bk1/4Bp1p/1p4p1/1P6/3p4/3P2PB/7q/1R3R1K w - - 0 1"] 1. Kxh2 h5 2. Qxf8+ Kh7 3. Rxf7# 1-0 [FEN "r1b1r1k1/p1q2p1p/1p2nQ2/5N2/3Rp3/6PB/P4P1P/6K1 w - - 0 1"] 1. Nh6+ Kf8 2. Qh8+ Ke7 3. Nf5# 1-0 [FEN "1r1q1k2/8/b1n1rb1N/2pp3P/pp4Q1/3P1NP1/PPP2P2/R5K1 w - - 0 1"] 1. Qg8+ Ke7 2. Qf7+ Kd6 3. Nf5# 1-0 [FEN "r1q4r/pp3n1p/2p1bN1k/2Pp2p1/1P3PP1/3R4/PQ2B2P/5RK1 w - - 0 1"] 1. Ng8+ Qxg8 2. Qf6+ Qg6 3. Rh3# 1-0 [FEN "r2r1knQ/pb4pB/1p2pp2/4R1B1/3q4/8/P4PPP/4R1K1 w - - 0 1"] 1. Qxg8+ Ke7 2. Qxe6+ Kf8 3. Qg8# 1-0 [FEN "r4nk1/3brp2/3p2p1/p1pPp1q1/2B1P2R/8/PP3P1Q/1K5R w - - 0 1"] 1. Rh8+ Kg7 2. Rg8+ Kf6 3. Qh8# 1-0 [FEN "8/8/6R1/pprkpN2/7p/2P4P/P4KP1/8 w - - 0 1"] 1. Ne3+ Ke4 2. Ke2 Rxc3 3. Rg4# 1-0 [FEN "r2qrbk1/6p1/2bPR2p/2p2B1Q/PnN5/2B3P1/7P/4R1K1 w - - 0 1"] 1. Qg6 Rxe6 2. Bxe6+ Kh8 3. Qxh6# 1-0 [FEN "8/5kbp/5pp1/2Br4/2Q5/6P1/5P1P/q3N1K1 w - - 0 1"] 1. Qxd5+ Ke8 2. Qe6+ Kd8 3. Bb6# 1-0 [FEN "3Q4/2K1R3/5k2/2P5/8/8/8/8 w - - 0 1"] 1. Qf8+ Kg6 2. Rg7+ Kh6 3. Qh8# 1-0 [FEN "4R3/4R2p/1p4p1/1P1p2k1/3b2P1/6K1/3rBr1P/8 w - - 0 1"] 1. h4+ Kf6 2. g5+ Kf5 3. Bg4# 1-0 [FEN "4nR2/p4R1p/2q3pk/4p3/4N1P1/3P4/7P/7K w - - 0 1"] 1. g5+ Kh5 2. Rxh7+ Kg4 3. h3# 1-0 [FEN "8/8/6r1/4Q3/4KPk1/p6p/7P/8 w - - 0 1"] 1. Qf5+ Kh4 2. Qxg6 a2 3. Qg5# 1-0 [FEN "r4r2/1p1R4/2p2k2/6NP/1PP2PP1/2b1N3/4P3/6K1 w - - 0 1"] 1. Ne4+ Ke6 2. Nc5+ Kf6 3. g5# 1-0 [FEN "rn1kb1R1/pp2q3/3p3B/1NpPpBQ1/2P5/7P/PP5K/8 w - - 0 1"] 1. Qxe7+ Kxe7 2. Bg5+ Kf7 3. Be6# 1-0 [FEN "2Q1bnk1/6pp/4pP2/2Bp4/1p3P2/8/1P5P/6K1 w - - 0 1"] 1. Qxe8 h5 2. Qxf8+ Kh7 3. Qxg7# 1-0 [FEN "2r1r2k/R4R2/1p5B/3N4/P3n3/8/2p3PP/6K1 w - - 0 1"] 1. Rh7+ Kg8 2. Rag7+ Kf8 3. Rg6# 1-0
304
Survey 4 Combinations (PGN)
[FEN "3K4/3N4/3p4/3k4/3B4/8/8/6Q1 w - - 0 1"] 1. Qd1 Kc6 2. Qb3 d5 3. Qb6# 1-0 [FEN "b1r1kb2/prp1ppp1/1pp1N3/8/8/4Q3/3R4/1n5K w - - 0 1"] 1. Qh3 fxe6 2. Qh5+ g6 3. Qxg6# 1-0 [FEN "Q4R1b/1r2pn2/2NpNPbR/3P1k2/BP6/3PK2P/p7/8 w - - 0 1"] 1. Bc2 Kxf6 2. Rxf7+ Kxf7 3. Qf8# 1-0 [FEN "2R3n1/5B2/5p1p/5p2/1p1PN3/3N4/4k2K/1bQ4n w - - 0 1"] 1. Rd8 Kxd3 2. Qd2+ Kxe4 3. Bd5# 1-0 [FEN "8/8/K2nn3/2k5/pN6/P2P1N2/1B6/8 w - - 0 1"] 1. Bd4+ Nxd4 2. Ng5 Nc8 3. Ne4# 1-0 [FEN "2b5/1p1p4/1P1K1p2/p1p1pQ2/k3N3/8/2P5/2B5 w - - 0 1"] 1. Qe6 Kb5 2. Qb3+ Ka6 3. Nxc5# 1-0 [FEN "3b4/4pK2/p2rppN1/r2k1Pp1/B1Rpp3/1P6/p2NP3/4Q3 w - - 0 1"] 1. Nb1 Rc6 2. Qb4 Rxc4 3. bxc4# 1-0 [FEN "q6k/6p1/4N1P1/8/8/B7/p3P3/K2Q4 w - - 0 1"] 1. Qf1 Kg8 2. Qf7+ Kh8 3. Qxg7# 1-0 [FEN "2K5/8/P3BB2/4P3/3ppkN1/2P2r2/5QRP/2b5 w - - 0 1"] 1. Bd8 dxc3 2. Qa7 e3 3. Qd4# 1-0 [FEN "8/7B/3p1p2/B3k3/1N1p1qQ1/n3pPn1/1P1N2K1/8 w - - 0 1"] 1. Qg6 Qxf3+ 2. Nxf3+ Ke6 3. Qe8# 1-0 [FEN "1Q5N/8/2k5/8/8/1N6/4BKb1/8 w - - 0 1"] 1. Ng6 Kd7 2. Bg4+ Kc6 3. Ne7# 1-0 [FEN "1B6/4bKp1/pnQ3B1/P2N4/1RPkrPRr/2p1p3/q1p2P2/2N2b2 w - - 0 1"] 1. Nc7 Rxf4+ 2. Qf6+ Kc5 3. Nxa6# 1-0 [FEN "8/8/8/8/3Q4/2pp2K1/n3k3/N6N w - - 0 1"] 1. Kf4 Kd1 2. Qxd3+ Kc1 3. Qc2# 1-0 [FEN "1rr1nQb1/1n6/pP1Pp3/1pPNb3/1pk1N1BR/1p2PP2/1P6/3R3K w - - 0 1"] 1. d7 Rxc5 2. Bf5 Rxd5 3. Nd2# 1-0 [FEN "K4Q2/p2BP1p1/3Rp2b/3pk1PP/3RN3/1P2p2b/1P2P3/8 w - - 0 1"] 1. Ra6 Kxd4 2. Qf4 dxe4 3. Qd6# 1-0 [FEN "8/3p4/2bB4/P1Qp4/k7/1r1B4/4nK2/n7 w - - 0 1"] 1. Ke1 Rb1+ 2. Bxb1 Nc2+ 3. Bxc2# 1-0 [FEN "4b3/4K2p/6pr/pP1kp3/7p/n1pBQR2/n4P2/8 w - - 0 1"] 1. Rf4 exf4 2. Qe6+ Kc5 3. Qd6# 1-0 [FEN "8/5N2/2k5/4p3/1B6/K2Q4/8/8 w - - 0 1"] 1. Qh7 Kb7 2. Nd6+ Ka8 3. Qb7# 1-0 [FEN "8/8/1Nn5/1p5p/3k1pP1/Q7/1p6/1K2R3 w - - 0 1"] 1. Nd7 Kc4 2. Re4+ Kd5 3. Nf6# 1-0 [FEN "2B5/1p2N3/1K1p4/3pp3/4k3/5p2/2R2P2/B3N3 w - - 0 1"] 1. Bd4 Kxd4 2. Bxb7 Ke4 3. Rc4# 1-0
305
1.4 Control Questions
Control questions were necessary to help filter out respondents who did not have a
reasonable understanding of the game or a justifiable taste for beauty in it. Given the
complexity of the surveys and effort required to take them properly, these additional
questions had to be kept simple yet provide a good indication if the respondent’s ratings
were believable. For this purpose, two pairs of highly contrasted chess combinations
were used and respondents were asked to select the more beautiful one in each pair.
Both questions had to be answered correctly. It was initially thought that the declared
chess ratings of respondents would help to filter out incompetent players, but in the end,
most simply claimed to be unrated (which is understandable).
In survey 1, the first pair consisted of the last three moves of the ‘Anderssen, A. vs.
Kieseritzky, L., London, 1851’ or ‘Immortal Game’ which is widely regarded as a gem
for the beauty of the winning combination (Burgess et al., 2004). This was paired
against ‘Rey, V. vs. Aragon, S., Ch Sub14 Femenino, 2004’ which was randomly
selected from a collection of games (Mega Database, 2008) between players with a low
Elo rating (i.e. 1300 each). It stands to reason that competent chess players would find
the former more beautiful (see the following subsection, Figure F.1). These
combinations were not identified by their names in the survey but for the reader’s
reference, the first one mentioned for each pair is the one on the left.
The second pair consisted of a famous composition by Al-Adli from the 9th century
(Levitt and Friedgood, 2008) contrasted against a composition which took the author
under 2 minutes to compose (compositions widely regarded as ‘ugly’ or ‘horrible’ in
chess literature are scarce). It also stands to reason that the former, which has been
306
admired for over a thousand years, should be the choice of a competent player (see the
following subsection, Figure F.2). Despite some modifications over the last thousand
years, the rules of chess have remained unchanged for the pieces involved in the Al-
Adli composition. Therefore, it can still be used for comparative purposes. Since the
first survey consisted of both tournament game combinations and compositions, the
control questions were also from both domains.
For survey 2, the first pair was the last three moves of the ‘Anderssen, A. vs. Dufresne,
J., Berlin, 1852’ or ‘Evergreen Game’. This was compared against ‘Emde, D. (Elo
1343) vs. Klauke, C. (Elo 1313), Hochsauerland-ch U18, 2003’. The second pair
compared a famous and beautiful composition, ‘Gurvich, A. S., Bakinski Rabochi,
1927’ (Levitt and Friedgood, 2008) against another quick composition by the author
himself. This survey also consisted of combinations from both domains.
For survey 3, both tournament game pairs (as used in survey 1 and survey 2) were the
control questions since the survey featured only tournament game combinations. In
survey 4, both composition pairs (as used in survey 1 and survey 2) were the control
questions. For each pair in all the surveys, respondents were asked, ‘Which
Combination Do You Think is More Beautiful?’ Incidentally (and this was not pre-
planned or pertinent to the survey), the computational aesthetic rating (based on the
proposed model) for the justifiably more beautiful combination was always higher.
307
1.4.1 Survey 1
1. Nxg7+ Kd8 2. Qf6+ Nxf6 3. Be7# 1. Qh6 Qxf2+ 2. Rxf2 g5 3. Qg7#
(A) (B)
Figure F.1 Survey 1: Control Question 1
1. Nh5+ Rxh5 2. Rxg6+ Kxg6 3. Re6# 1. Qb6+ Ka8 2. Bd5+ Rc6 3. Bxc6#
(A) (B)
Figure F.2 Survey 1: Control Question 2
308
1.4.2 Survey 2
1. Bf5+ Ke8 2. Bd7+ Kf8 3. Bxe7# 1. Nxe5+ Ke8 2. Qd7+ Kf8 3. Qf7#
(A) (B)
Figure F.3 Survey 2: Control Question 1
1. Ng3+ Kg1 2. Ng5 Nhg4 3. Nf3# 1. Rh5 a3 2. Ra5 Kc8 3. Ra8#
(A) (B)
Figure F.4 Survey 2: Control Question 2
309
1.4.3 Survey 3
1. Nxg7+ Kd8 2. Qf6+ Nxf6 3. Be7# 1. Qh6 Qxf2+ 2. Rxf2 g5 3. Qg7#
(A) (B)
Figure F.5 Survey 3: Control Question 1
1. Bf5+ Ke8 2. Bd7+ Kf8 3. Bxe7# 1. Nxe5+ Ke8 2. Qd7+ Kf8 3. Qf7
(A) (B)
Figure F.6 Survey 3: Control Question 2
310
1.4.4 Survey 4
1. Ng3+ Kg1 2. Ng5 Nhg4 3. Nf3# 1. Rh5 a3 2. Ra5 Kc8 3. Ra8#
(A) (B)
Figure F.7 Survey 4: Control Question 1
1. Nh5+ Rxh5 2. Rxg6+ Kxg6 3. Re6# 1. Qb6+ Ka8 2. Bd5+ Rc6 3. Bxc6#
(A) (B)
Figure F.8 Survey 4: Control Question 2
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1.5 Respondent Ratings
This section lists the actual valid respondent (R) ratings for each combination in each of
the surveys. The combination numbers (C) correspond to those in Appendix F, section
1.2. This information is provided here for the benefit of other researchers who might
wish to use these combinations and their aesthetic ratings for other experiments. The
average respondent ratings for the combinations used in all four surveys are available in
chapter 6; Tables 6.10, 6.11, 6.13 and 6.14, respectively.
Survey 1 Respondent Ratings:
R/C 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2.0 3.0 5.0 7.0 8.0 4.0 3.0 4.0 5.0 7.0 2.0 8.0 8.0 7.0 5.0 3.0 7.0 9.0 6.0 8.0
2 1.0 6.9 10.0 9.4 4.0 3.9 2.9 2.5 3.5 4.5 1.5 6.0 1.5 5.0 7.0 6.9 5.0 7.9 4.0 9.2 3 7.5 3.5 3.0 2.2 5.7 6.0 4.0 8.0 5.0 5.5 6.5 6.7 3.7 7.7 2.0 3.7 2.5 4.2 7.0 1.0 4 5.5 6.0 6.0 6.1 4.2 6.6 5.2 7.0 7.8 3.7 3.5 8.0 6.6 6.2 4.0 8.5 3.2 6.7 7.4 5.8 5 1.5 4.5 6.5 7.5 6.0 6.7 2.0 4.8 3.9 2.0 1.0 2.5 8.0 1.0 4.0 3.2 3.0 7.3 1.0 5.0 6 4.0 5.8 7.5 7.0 6.0 7.2 7.3 5.4 7.8 8.5 3.0 4.5 5.4 7.3 4.3 6.7 6.4 7.1 5.7 5.4 7 4.1 1.0 1.0 5.5 2.1 3.5 6.4 7.7 9.3 2.1 1.0 1.5 5.8 2.2 6.3 9.5 1.0 9.8 9.0 2.3 8 1.1 5.0 3.0 8.0 3.5 4.0 5.5 1.5 2.0 8.5 1.0 2.5 2.4 8.5 2.3 6.0 2.2 4.5 2.1 1.9 9 1.0 3.0 2.0 4.0 1.0 2.0 5.0 3.0 3.5 4.0 3.0 4.5 5.0 6.0 4.0 5.5 3.0 4.0 2.5 1.5
10 4.1 4.0 8.0 9.9 2.1 3.2 3.5 1.0 6.6 3.0 1.0 9.5 8.5 7.0 5.9 2.2 5.0 3.4 1.7 1.2 11 5.2 6.1 6.2 7.1 5.5 5.6 5.1 6.8 7.1 8.1 6.2 8.1 7.8 8.5 8.4 7.4 6.9 7.2 5.9 6.3 12 3.0 2.5 5.0 6.0 6.1 6.0 6.5 5.0 6.7 4.0 2.0 5.1 5.5 5.8 4.5 6.0 6.5 7.0 7.5 5.0 13 2.0 2.5 6.0 5.5 5.0 3.5 5.5 4.0 3.0 5.5 3.5 5.5 5.5 4.5 5.0 3.5 5.0 3.0 2.0 3.5 14 7.5 2.5 3.2 3.0 4.2 3.8 2.0 6.5 6.0 7.4 4.2 5.5 6.2 7.5 5.8 6.5 4.5 7.0 6.2 5.8 15 5.2 6.3 7.0 6.6 7.9 7.1 8.2 9.0 4.5 7.7 2.0 4.1 7.8 6.5 8.3 7.9 5.4 8.1 8.8 8.6 16 5.7 6.3 7.3 8.3 8.9 7.5 7.6 5.3 8.2 6.2 4.6 9.2 8.2 9.5 3.2 5.1 7.3 3.5 5.4 6.3 17 1.3 2.2 1.4 5.5 1.6 3.4 2.7 1.1 2.9 4.4 1.2 6.3 2.5 2.9 1.2 2.3 2.2 3.3 1.3 1.5 18 9.8 8.5 10.0 10.0 8.5 8.3 9.0 6.2 8.1 8.6 7.5 8.4 8.8 8.1 7.6 7.5 8.4 9.1 8.0 8.7 19 3.0 4.7 5.4 7.8 4.4 4.6 7.2 2.0 4.2 6.2 2.2 6.8 4.0 5.4 1.5 3.8 4.8 5.6 2.8 2.9 20 2.5 4.0 6.8 5.5 1.8 4.7 5.2 1.3 6.1 3.5 1.0 2.9 3.3 3.9 0.7 5.6 2.2 3.2 2.4 1.6 21 3.5 2.1 3.7 5.2 4.4 3.3 5.7 6.1 6.2 7.0 3.4 2.1 4.6 3.6 7.9 8.2 7.5 8.0 7.4 7.2 22 1.6 1.0 1.2 9.6 2.6 7.9 10.0 1.0 2.1 5.1 1.4 6.2 3.4 7.9 4.0 5.7 6.9 1.7 1.0 2.0 23 3.0 5.0 7.0 9.0 5.0 8.0 5.0 3.0 7.5 8.0 3.0 8.0 8.5 9.0 8.0 7.0 2.0 7.5 5.0 2.0 24 5.0 7.0 6.0 4.0 6.0 5.0 4.0 6.0 7.0 6.5 4.0 6.0 5.7 4.8 5.0 4.0 4.7 4.9 5.6 4.5 25 4.5 7.0 8.5 9.5 1.0 3.0 7.5 1.0 5.5 9.5 1.0 6.5 5.5 9.0 4.0 7.0 5.5 2.5 3.5 5.0 26 4.5 5.5 7.8 9.7 3.3 5.2 7.7 5.5 6.2 8.9 5.3 7.9 6.1 2.9 7.5 4.2 5.2 3.3 3.7 4.3 27 4.3 3.1 2.1 8.2 5.7 6.0 8.2 5.4 6.2 1.7 3.2 5.6 7.2 8.2 9.3 4.2 7.0 6.9 3.4 5.0 28 3.6 6.0 4.8 8.7 2.0 4.0 3.7 5.3 7.7 5.5 6.0 8.3 8.5 8.4 2.0 9.3 6.7 7.7 8.2 8.2 29 2.5 1.0 3.5 4.0 3.0 1.5 2.5 4.5 3.5 3.0 1.0 2.5 3.0 4.5 5.0 4.5 1.0 3.0 4.5 2.0 30 5.0 5.5 6.0 6.2 6.5 6.5 6.5 6.0 7.0 6.8 6.0 6.8 6.5 6.0 6.0 6.5 6.2 6.6 6.9 6.8 31 3.0 2.5 2.0 2.0 4.0 5.0 1.8 5.5 5.6 2.7 1.0 4.8 5.2 3.8 0.8 6.0 5.8 7.0 4.7 3.8 32 5.0 4.0 7.0 5.0 3.0 5.0 6.0 5.0 4.0 7.0 3.0 4.0 5.0 2.0 4.0 6.0 2.0 5.0 5.0 4.0
312
R/C 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 33 1.3 2.1 2.2 3.4 3.6 1.5 1.2 1.1 1.4 3.7 3.1 4.4 2.3 5.7 5.4 5.7 6.3 2.3 2.8 3.1 34 4.0 7.0 7.0 2.5 5.0 4.0 8.0 1.0 6.5 8.0 5.5 10.0 3.0 6.5 8.0 4.5 7.0 4.0 4.0 3.0 35 3.0 2.5 5.0 4.0 1.5 3.0 1.5 3.0 3.5 1.5 2.0 2.5 3.0 6.0 2.0 4.5 4.0 5.5 3.5 5.0 36 4.1 3.6 5.2 6.7 4.8 4.5 3.0 3.0 6.1 7.2 1.2 6.8 5.4 7.7 4.8 5.1 3.6 3.9 1.3 5.5 37 5.0 6.0 6.7 6.5 3.0 6.8 3.9 7.0 6.9 6.3 2.0 2.3 7.5 6.0 5.8 6.9 6.5 6.2 6.7 7.0 38 1.5 2.5 2.1 1.6 1.8 2.2 1.9 1.3 2.3 3.3 1.4 4.1 1.8 2.5 3.0 2.4 1.6 1.9 1.2 1.3 39 3.0 5.0 6.5 7.0 6.0 5.0 6.5 6.0 7.0 9.0 4.5 5.0 5.0 8.5 7.5 6.0 5.0 5.0 5.0 5.0 40 3.0 3.0 2.5 8.0 3.0 4.0 4.0 1.0 4.0 7.0 2.5 3.0 3.0 4.0 1.0 4.5 3.0 8.0 6.0 4.0 41 2.0 1.0 1.5 2.6 4.2 3.3 3.6 3.8 3.7 3.0 3.5 4.0 5.0 3.0 7.0 5.5 3.4 6.5 4.8 6.0 42 5.1 6.1 6.2 7.4 7.7 5.8 8.2 5.2 5.4 6.0 4.8 5.4 3.8 5.2 3.6 4.0 3.8 4.2 3.0 4.8 43 3.2 6.2 6.2 5.7 4.4 6.6 6.9 4.1 7.3 7.0 2.5 6.9 5.8 8.2 1.0 7.5 3.8 7.1 8.0 5.2 44 3.0 2.0 5.0 5.0 3.5 4.0 3.2 4.0 4.0 4.5 0.5 4.0 4.5 4.5 4.3 4.0 3.0 1.0 2.0 1.5 45 3.5 4.2 3.0 4.5 2.5 3.5 4.8 3.1 3.5 5.5 2.6 4.4 5.0 5.5 5.2 5.8 4.8 6.9 3.3 4.9 46 5.2 1.9 8.2 3.7 4.5 7.6 4.1 5.0 7.8 1.1 2.3 7.4 6.1 3.6 7.8 8.8 4.4 8.3 5.1 6.9 47 5.6 7.2 8.5 7.8 7.5 7.5 7.5 5.9 7.2 8.2 5.5 6.4 6.2 6.2 5.5 6.5 5.6 5.8 5.8 5.5 48 2.0 6.0 7.0 8.0 2.0 4.0 5.0 3.0 5.0 8.0 1.0 9.0 4.0 7.0 1.0 5.0 4.0 8.0 3.0 4.0 49 2.5 6.3 4.2 3.8 1.1 2.1 1.2 1.8 2.8 4.6 1.3 3.1 3.6 4.4 1.2 2.3 2.7 3.7 1.5 1.2 50 5.0 7.0 4.0 7.1 3.0 4.5 5.2 4.3 6.5 2.1 2.0 2.5 5.3 8.0 3.2 6.4 2.2 5.6 6.1 4.4 51 5.0 6.0 7.0 5.0 5.0 6.0 6.0 5.0 6.0 7.0 5.0 6.0 6.0 6.0 6.0 6.0 5.0 6.0 5.0 5.0 52 4.8 3.1 4.0 5.0 6.2 4.8 6.8 7.2 7.5 8.7 6.5 7.0 6.8 9.0 7.7 9.2 6.1 8.9 7.0 7.5 53 5.0 8.3 9.2 9.0 9.5 7.3 8.0 8.5 6.2 8.4 4.2 4.8 6.5 8.3 8.1 9.2 4.0 7.1 5.2 7.0 54 1.5 6.5 8.5 2.5 7.0 3.0 8.0 1.0 1.0 7.0 1.5 6.5 7.0 9.5 6.5 4.0 1.0 4.0 2.0 3.0 55 3.0 3.0 7.5 6.5 5.0 6.5 4.0 2.5 5.0 7.0 3.0 6.5 5.5 9.0 3.5 7.5 4.0 6.0 3.5 3.0 56 5.0 7.0 8.0 9.0 4.0 7.0 4.0 4.0 7.0 3.0 4.0 4.0 5.0 4.0 2.0 4.0 4.0 4.0 3.0 2.0 57 1.5 1.6 3.5 6.2 5.9 4.3 3.2 2.7 3.3 3.7 0.8 6.4 3.6 4.4 3.5 4.7 2.0 3.0 1.1 2.7 58 6.3 7.4 7.0 4.6 7.8 8.9 10.0 1.0 8.2 9.4 3.2 8.8 7.9 5.5 4.7 6.8 6.2 9.5 4.4 5.6 59 2.0 9.0 5.0 6.5 3.0 3.0 1.0 0.5 2.0 3.0 0.5 5.0 3.0 0.5 0.5 3.0 2.0 5.0 2.5 1.5 60 3.0 3.0 4.5 4.0 3.5 4.3 5.1 2.5 1.2 7.3 8.1 7.3 7.5 8.2 4.5 8.3 3.8 2.1 5.5 7.5 61 3.1 8.4 9.3 7.1 6.1 4.9 6.0 4.4 4.7 7.6 1.7 4.2 2.8 5.6 6.9 5.0 4.1 5.2 5.3 3.5 62 3.0 5.0 6.7 7.4 4.1 3.6 7.2 2.6 8.2 8.4 3.7 6.8 2.3 4.4 2.1 3.0 2.5 3.7 4.1 3.3 63 1.4 4.5 2.2 6.2 5.4 3.2 7.8 1.2 2.4 9.2 0.7 2.0 2.4 5.8 1.5 5.0 2.9 3.4 1.8 1.2 64 4.0 4.5 6.0 6.0 6.5 7.0 7.5 7.0 7.0 5.0 5.0 6.5 8.0 7.5 8.0 7.0 5.0 7.5 7.0 7.5 65 6.0 8.2 8.4 9.0 9.0 9.0 6.0 4.0 3.8 6.1 3.1 7.0 8.0 10.0 9.3 7.1 5.2 8.3 7.1 7.2 66 4.0 4.0 4.0 5.0 5.0 5.0 2.0 2.0 5.0 2.0 1.0 5.0 6.0 3.0 4.0 6.0 3.0 7.0 3.0 3.0 67 5.5 4.3 6.0 7.0 6.5 7.5 4.5 8.0 7.0 7.3 4.5 6.0 8.5 9.0 7.3 9.3 7.5 8.8 8.8 8.5 68 7.5 9.5 8.0 10.0 7.5 6.5 9.0 4.5 6.5 8.0 7.5 9.5 6.0 6.5 7.0 7.0 8.5 6.5 5.0 8.0 69 6.5 7.5 8.5 9.5 6.0 8.0 8.5 6.5 5.5 7.5 4.0 9.0 6.5 6.5 6.0 5.0 9.0 8.5 8.0 7.5 70 2.5 9.0 4.3 7.3 3.7 4.5 6.4 2.3 4.3 5.9 1.9 8.1 3.1 6.3 4.5 4.3 4.6 3.4 2.1 2.5 71 5.7 6.2 8.9 3.2 1.2 9.8 6.3 3.6 4.2 2.4 8.5 7.2 6.5 4.2 6.3 7.8 9.9 9.1 4.6 2.9 72 2.0 5.2 5.4 6.1 4.8 5.0 4.9 1.5 2.4 6.3 3.1 3.8 3.0 3.7 2.4 3.8 3.0 2.8 2.9 3.2 73 5.0 6.5 6.0 4.0 3.0 6.2 2.1 6.1 7.0 8.0 3.9 8.1 8.5 8.5 3.0 9.0 9.1 7.0 8.0 7.5 74 2.9 4.3 6.4 7.5 3.7 1.9 5.4 3.8 6.2 6.9 2.4 7.2 6.2 8.2 4.7 6.4 2.4 5.6 3.7 6.7 75 1.0 2.0 5.0 6.0 2.0 2.0 3.0 1.0 4.0 8.0 1.0 4.0 3.0 5.0 1.0 3.0 4.0 6.0 3.0 2.0 76 5.2 4.5 9.0 2.0 5.9 6.0 5.0 5.3 6.3 8.3 4.0 8.8 6.1 5.5 4.0 4.8 4.6 5.0 3.0 5.5 77 5.1 2.2 4.7 5.9 4.8 5.7 3.3 5.0 5.5 4.7 3.9 4.2 6.1 6.2 5.2 5.9 4.4 5.4 4.0 5.6 78 4.0 3.0 8.0 7.0 7.0 3.0 6.0 9.0 7.0 6.0 2.0 7.5 8.0 8.0 9.0 8.5 6.0 7.0 6.5 4.0 79 6.0 2.5 6.5 8.0 4.5 6.0 7.0 6.5 5.0 5.0 7.0 6.3 7.2 6.7 8.5 7.6 4.5 7.2 6.0 5.8 80 3.0 4.3 4.5 4.7 4.6 4.5 4.4 4.0 4.6 2.9 2.7 3.4 4.0 4.7 1.2 4.7 2.7 4.8 1.3 1.5 81 1.0 6.0 7.5 8.0 4.0 4.0 8.0 2.0 2.0 8.0 1.0 5.0 2.0 4.0 2.0 2.0 4.5 1.0 0.5 3.0 82 8.0 7.0 9.0 9.8 5.5 9.9 10.0 2.2 8.5 7.9 2.1 3.5 4.5 9.9 1.0 6.5 3.5 9.1 3.1 1.1 83 3.0 2.0 6.0 8.0 5.0 8.0 5.0 8.0 4.0 3.0 4.0 7.0 9.0 9.0 7.0 8.0 6.0 8.0 5.0 5.0 84 6.0 1.0 5.4 7.2 4.0 3.0 2.0 3.4 4.0 6.5 2.0 4.8 7.0 5.4 4.0 6.4 1.4 3.0 3.2 5.0
313
R/C 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 85 4.0 3.0 4.0 6.0 4.0 5.0 4.5 5.5 6.0 5.0 3.0 6.0 7.0 6.0 5.0 6.0 4.0 7.0 5.0 6.0 86 5.5 6.9 7 7.5 6.7 6.8 6 7.1 5.8 6.7 4.5 5.2 6.8 7.8 7.1 7.3 6.4 6.9 5.7 6.1 87 6.3 3.5 6.5 9.0 8.3 6.0 4.5 5.0 6.8 8.0 2.0 8.5 7.5 8.0 6.8 6.0 4.5 7.3 2.5 4.0 88 5.0 7.2 8.0 9.9 2.0 4.5 6.0 3.8 7.1 9.2 6.3 8.1 9.4 3.2 1.6 8.8 5.3 6.9 4.9 9.3 89 4.5 6.8 7.5 9.0 5.5 6.5 7.5 8.0 7.5 6.5 3.5 9.5 7.0 8.0 8.5 8.0 6.0 8.0 7.0 7.5 90 2.0 4.0 6.0 7.0 5.0 4.0 7.0 3.0 5.0 7.0 1.0 7.0 6.0 3.0 3.0 4.0 7.0 6.0 4.0 6.0 91 8.0 6.5 3.0 8.5 8.0 7.0 9.0 6.0 8.5 5.0 1.5 8.5 7.5 4.5 8.5 8.0 4.0 5.5 6.0 7.0 92 2.0 1.0 5.0 6.0 7.0 7.0 6.0 8.0 8.0 6.0 2.0 7.0 9.0 9.0 10.0 10.0 8.0 9.0 9.0 9.0 93 5.0 2.0 7.0 4.0 8.0 7.0 4.0 5.0 5.0 4.0 2.0 4.5 6.0 6.5 7.0 5.0 3.0 4.0 5.0 5.0 94 2.5 3.1 3.0 7.0 3.0 4.0 7.5 3.0 4.1 4.0 2.0 6.5 3.5 3.8 4.3 4.8 4.6 5.2 3.8 3.8 95 5.0 4.5 6.0 4.4 3.3 5.5 4.0 6.0 3.4 4.5 2.5 4.0 5.4 6.0 1.5 6.0 3.8 5.0 6.6 2.3 96 5.3 6.4 8.9 7.4 6.3 6.7 7.4 2.4 3.5 7.3 2.7 7.1 4.2 5.2 3.9 2.6 6.4 4.1 2.1 2.3 97 3.0 5.0 4.0 6.0 4.0 4.0 5.0 2.0 5.0 6.0 3.0 7.0 5.0 4.0 5.0 5.0 4.0 6.0 3.0 4.0 98 4.0 6.5 8.5 7.7 7.5 7.2 6.5 5.0 6.0 8.5 7.3 7.5 6.7 7.0 5.8 6.0 7.0 5.5 5.4 6.0 99 5.0 7.5 8.0 8.8 9.3 6.5 6.0 5.0 7.5 6.0 4.5 6.5 5.5 5.0 3.5 5.5 3.0 5.5 6.5 6.0
100 3.5 3.2 4.5 5.5 2.3 3.7 4.6 2.1 3.6 4.5 2.1 6.2 6.1 3.4 6.4 5.9 4.2 2.1 3.2 4.2 101 4.1 6.0 5.0 7.0 2.2 5.1 1.0 1.5 1.6 6.1 0.0 5.5 1.7 8.0 8.1 7.0 5.4 7.9 6.3 8.1 102 1.0 9.0 2.0 5.0 1.5 5.5 3.0 1.0 1.9 6.0 1.0 4.0 3.0 5.0 5.0 1.8 6.7 5.0 1.0 2.3 103 4.0 5.5 4.5 6.5 3.0 6.0 3.0 3.0 4.0 6.5 3.5 7.5 6.0 8.0 4.0 7.0 3.5 6.5 5.0 4.5 104 3.0 5.5 6.5 4.2 5.3 5.8 6.4 3.2 4.0 6.1 1.5 4.3 2.6 4.6 2.5 1.8 2.7 3.2 1.0 2.8 105 2.1 5.5 7.3 8.5 8.7 1.7 6.8 2.3 1.9 9.0 1.1 7.2 2.5 7.9 2.8 2.5 7.0 2.0 1.4 1.7 106 6.0 1.0 1.5 1.3 1.5 2.0 1.7 5.0 6.5 1.5 2.2 7.0 6.8 6.5 6.2 6.5 6.2 6.5 6.5 6.5 107 2.0 3.0 2.5 2.5 2.0 2.0 3.0 1.0 3.0 4.0 1.0 4.0 1.0 2.0 1.0 2.0 3.0 3.0 2.0 2.0 108 6.5 9.5 8.5 10.0 9.0 8.5 10.0 5.5 7.0 9.5 8.0 9.5 6.5 9.0 7.5 7.5 9.0 6.0 3.5 7.0 109 6.0 8.0 8.5 9.0 7.0 7.0 7.5 6.0 7.5 8.5 3.0 7.5 7.5 8.0 2.0 8.0 5.0 7.5 6.0 7.5 110 3.4 5.4 4.2 6.7 4.1 4.8 9.1 4.2 5.1 3.7 1.9 9.4 7.4 8.1 10.0 5.1 4.8 3.1 3.9 6.8 111 7.0 6.0 4.0 10.0 6.5 7.0 6.0 2.0 7.0 4.0 5.0 6.0 6.0 2.0 8.0 8.5 1.0 4.0 2.0 3.0 112 5.0 3.2 6.5 7.0 6.0 7.0 7.2 6.0 6.5 3.0 4.0 4.0 6.0 7.2 7.0 8.0 5.0 5.0 5.0 6.0 113 7.8 8.1 6.7 7.1 8.5 8.7 9.2 8.0 10.0 6.5 9.4 9.2 8.9 10.0 7.9 9.8 7.7 8.5 8.1 8.2 114 3.1 5.4 5.9 7.5 4.5 6.5 5.2 2.9 5.4 8.1 6.5 6.3 6.1 6.3 7.5 5.3 6.8 6.3 7.3 5.6 115 1.5 8.5 4.0 3.5 1.3 1.3 2.5 1.1 1.2 4.0 1.0 6.7 1.4 3.2 1.3 1.2 5.6 1.4 1.5 1.6 116 1.0 6.0 7.0 9.0 2.0 6.5 4.5 4.0 4.0 8.0 1.0 8.5 1.5 7.5 6.0 5.0 8.0 5.5 2.0 2.0
Survey 2 Respondent Ratings:
R/C 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 5 3 3 6 4 7 6 8 6 8 7 8 6 4 8 3 7 8 8 6 2 6 5 8 3 4 2 1 3 5 5 6 8 7 9 9 5 1 4 8 4 3 6 2 6 5 2 5 3 4 6 5 3 6 5 7 7 8 7 8 6 7 4 9 6 9 6 6 8 5 6 7 8 5 3 3 10 3 9 3 8 5 4 5 8 6 6 7 5 7 5 6 7 6 5 7 8 7 7 6 5 7 5 7 6 5 1 7 3 2 3 2 2 4 5 2 3 5 6 6 7 6 5 9 6 7 2 3 4 5 5 6 6 7 8 7 7 9 8 9 9 8 8 9 10 9 8 6 2 7 2 1 6 2 3 6 3 2 8 7 8 6 5 1 7 9 10 9 6 6.5 7 7 2 6 1 2 4 5.5 6 6.5 8 8.5 7.5 8.5 7.5 8.5 8 10 10 8 3 9 7.5 5.5 8.5 7 6 8 8.5 8 9.5 9 9.5 10 8.5 10 9.5 9.9 10 11 8.5 3 8 5.5 5 7.5 3 4 7 6 7 5.5 5 8 6 7.5 5 7 5 6 12 9 6 8.5 3 4 7 2.5 4 6.5 5.5 3.5 4.5 5 6.5 6 8.5 7 7.5 5 4.5 13 2 2 3 4 2 4 2 4 2 4 3 5 5 5 7 5 6 4 6 4 14 1.5 4.5 2 4 1 6.5 5 3.5 7 7.5 6 8.5 8 6.5 9 8 10 9.5 8.5 9.5
314
R/C 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 15 3 1 8 4 2 3 2 1 3 3 1 2 2 7 3 7 3 3 6 2 16 6 2 7 3 1 4 3 4 5 4 6 1 4 7 7 6 8 9 9 8 17 5 2 8 6 2 8.5 2 1.5 4 3.5 3 6 4 8 3.5 5 4.5 4 4.5 6 18 8 5 6 4 3 3 3 3 4 4 3 4 3 6 5 5 4 3 3 5 19 9 4 9 7 6 7 6 5 6.5 8 7.5 7 8.5 9 7.5 8 9.5 7 10 7 20 5 4 6 5 4 3 5 6 7 4 5 7 6 8 6 7 5 3 9 4 21 7 4 7 6 6.5 6 5.5 6 7 6.5 4 7.5 6.5 7 7.5 6.5 9.5 9 6.5 8 22 7 4 8 6 3 7 7 4 6 8 6 9 8 10 9 6 6 9 8 10 23 5.3 1.6 6.4 5 1.4 6.8 1.8 1.3 6.2 4.7 1.9 7 6.2 8.2 7.5 5.4 5.7 6 7 8.7 24 9 3 8 6 5 8 4 8 10 8 5 6 6 10 4 9 2 5 4 6 25 9 1 8 5 2 5 4 1 4 5 2 1 3 5 3 2 4 2 2 1 26 7 1 5 1.5 1 6.5 1 2 2 6.5 4 2 4 9 5 7.5 6 3.5 7 4.5 27 8 7 8 7 6 7 6 6 8 8 7 7 8 9 7 8 6 9 8 9 28 9.8 5.8 9.6 7.3 4 8.7 6.3 6.5 8.8 9.3 4.7 7.6 9.2 9.9 9.7 9.1 7.9 9.4 9.2 8.5 29 8 3 4 5 2 5.5 2 5 8 8 7 3 3 9 8 9 6.5 2 8 2 30 8 3 8.5 6 2 7.5 4 2 6 7 5 8 5 7 8 9 7 4.5 8 7.5 31 4 5 8 9 2 7 2 2 6 5 6 9 9 8 10 5 7 6 10 7 32 7 2 6 3 1 6 4 5 4 8 1 6 5 9 6 5 5 5.1 7 7.1 33 6 1 7 5 4 5 4 3 6 5 8 7 4 8 7 6 8 7 9 8 34 5 2 6 4 1 2 1 4 5 4 1 2 1 4 3 6 1 2 1 3 35 10 4 9 5 4 2 3 6 7 4 4 4 6 7 7 8 6 7 8 8 36 5 4 7 7 4 3 2 2 5 6 8 7 6 6 8 6 7 4 7 7 37 7 3 6 5 3 6 2 6 7 6 2 6 4 8 6 7 6 7 7 8 38 5 4 6 8.5 3 7 4 6 8 8.5 6.5 7 9 9.5 9.5 5 8.5 7 9 9.5 39 7 3 7 4 1 8 2 3 7 8 3 4 5 9 7 8 6 5 7 4 40 5 4 8 7 2 3 4 5 7 6 5 6 8 9 8 8 9 8 8 8 41 6 1 8 2 1 3 2 4 8 5 4 9 5 10 6 7 3 8 6 7 42 8 3 8.5 6.5 5 7 5 4 8.5 5.5 3 5 6 8.5 6 8.5 5 5.5 6 6.5 43 5 3.5 5.5 7 2 7 4 5 6.5 5.5 2 9 8 10 9.5 7 6 7 9 10 44 10 3 5 6 2 4 6 2 6 7 7 8 6 9 8 6 9 6 8 8 45 6 4 7 5 3 5 5 4 6 7 5 2 1 3 2 7 7 4 7 3 46 7 5 7 8 4 7 4 3 6 8 2 6 4 5 6 5 8 8 8 9 47 8 6 8 6 4 7 4 3 7 6 2 7 7 9 7 6 7 8 7 8 48 2 1 5 1 1 4 1 1 3 4 5 6 7 8 8 9 7 10 9 10 49 8 4 7 4 3 5 4 5 6 7 5 7 6 7 7 8 7 8 9 6 50 8.9 7.5 8.5 5 4.5 7 8 5.5 6.8 7.2 4 8.3 6.2 8.5 7.7 7.8 6.2 3.5 6.8 6.9 51 8 2.5 9 3 1.5 4.5 2.2 2.3 5.5 4.5 4 3.5 3 9 8 6.5 3 5.5 9 9.5 52 8 4 9 6 7 8 3 6 9 7 5 10 8 9 8 10 7 9 7 6 53 7 2 8 2 2 7 2 5 8 6 3 7 7 9 9 7 6 7 7 9 54 8 5 8 10 5 5 7 5 8 6 5 10 9 9 4 7 6 6 8 9 55 4.5 2 7.5 6.5 4.5 3.5 2 1.8 5 4 3 6 6 9 8.5 8.8 7.5 7.7 10 6.5 56 5.6 1 7.4 7.8 2 8.5 2.5 2.4 8.7 6.8 7.2 3.5 3.5 9 4 6.1 9.3 7 10 7.9 57 10 1 9 5 2 4 3 2 7 5 4 8 6 9 7 5 6 7 10 8 58 8 1 8 3 1 3 2 4 6 4 5 6 7 10 9 7 3 8 10 7 59 6 1 7 4 2 7 5 2 7 6 7 10 9 10 9 6 9 7 10 10 60 8 5 6 5 4 7 5 6 6 7 4 6 7 8 7 8 6 7 6 8 61 6 3 6 7 3 8 4 6 7 5 5 10 7 10 5 7 6 3 9 5 62 7.5 3.5 8 6 3 4 7 5.5 8.5 7.5 6.8 8.5 8 9 8.7 8.3 6.5 8.5 6 8 63 8 5 7 5 2 7 3 4 6 8 7 4 2 3 2 6 6 5 10 4 64 10 3 10 3 2 4 3 2 5 3 3 7 5 10 6 9 7 9 10 9 65 8 2 7 9 4 9 2 4 6 7 3 8 9 10 10 8 9 10 10 8 66 8 2 6 4 1 5 3 5 8 6 5 4 6 9 8 7 2 3 1 8 67 6 5 7 6 4 7 3 4 5 7 5 7 7 8 7 8 3 7 8 6
315
R/C 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 68 8 6 8 7 6 7 6 6.5 6 7 6 7 7 8 7.5 8 6 6.5 8 7.5 69 5 5 7 6 4 5 3 4 6 7 5 8 8 9 8 10 9 7 9 10 70 4 3 5 5 3 4 4 4 5 6 3 5 5 7 6 7 3 7 5 6 71 10 6 9 5 4 3 2 2 5 5 1 1 1 6 2 10 1 4 5 1 72 6 4 8 5 2 4 7 5 7 4 9 8 5 10 7 6 9 9 8 8 73 4 3 8 5 2 6 3 4 3 5 4 9 8 10 7 6 8 10 9 7 74 5 6 5 7 3 6 3 6 6 6 3 7 8 8 8 5 7 9 6 9 75 7 3 6 3 2 4 2 3 4 6 4 7 4 9 6 8 5 8 7 9 76 9 3 8 4 4 4 3 4 5 5 3 6 5 6 4 6 6 6 7 6 77 8 1 8 2 1 4 3 1 3 4 2 5 4 9 6 8 6 5 5 7 78 10 2 7 5 1 7 1 5 8 6 6 8 8 9 7 10 9 10 7 9 79 8.5 5 9 5 4 7 7 4.5 6.5 9.5 7.5 9.5 8.5 10 9 8 8.5 9 9.5 10 80 5 1 6 8 5 6 4 5 8 9 8 10 9 10 9 8 10 10 10 10 81 8 5.4 10 8.1 5.3 7.9 5.1 4.7 9.4 8.6 7.2 7.8 7.8 9.3 7.5 8.6 8.6 8.9 9.7 8.6 82 3 1 4 2 1 2 1 2.5 3 2 2.3 7 4 7.5 5 3 7 8 9 8.5 83 6 4.5 7.5 5.3 4.5 5.3 5 6 7 6.5 7 7.5 7.5 8.5 6.5 7.5 6.5 7.8 9 8 84 9 7 9 8 6 8 6 5 7 8 8 9 9 10 9 8 5 7 7 8 85 7 8 10 9 6 8 7 5 8 7 8 6 3 5 5 9 7 3 8 4 86 5 3 7 5 3 4 2 2 8 7 4 6 5 9 7 8 9 4 8 9 87 10 2 10 7 1 4 3 3 8 9 7 10 9 10 10 10 10 10 10 10 88 7 6 7 6 6 7 6 7 6 8 8 7 8 10 7 8 8 7 9 8 89 3 2 5 3 3 2 3 2 4 5 4 7 5 8.5 6 8.5 8.5 7 8.5 7 90 6 3 4 6 2 7 4 5 6 7 5 7 8 7 8 4 5 8 8 9 91 3 1 4 2 2 3 3 1 2 3 1 5 6 7 8 4 8 6 7 9 92 8 4 8 7 5 3 6 3 6 5 5 6 4 7 3 6 2 6 9 5 93 8 5 7 7 1 5 5 5 1 25 5 9 9 10 9 7 10 9 10 10 94 9 6 7 8 6 8 6 5 7 6 7 8 9 8 6 7 7 8 7 6 95 10 7 8 7 7 6 6.5 6 7 6 6.5 6 6 9 9 9 8 8 8 8.5 96 7 1 7 6 2 7 3 4 6 7 5 8 6 9 7 9 8 7 10 9 97 10 1 10 3 1 2 4 1 4 1 8 5 7 10 8 9 9 10 10 8 98 8 5 6 4 5 7 3 3 8 6 4 9 6 9 10 8 8 10 6 9 99 7 5 7 6 3 2 2 5 4 5 2 6 1 8 3 7 5 6 8 6
100 2 1 3 4 2 4 3 2 3 4 6 8 7 6 5 4 9 7 10 8 101 10 5 8 6 6 5 5 4 6 7 8 9 8 10 10 9 5 6 8 7 102 5 4 6 6 5 4 4 4 6 5 7 2 2 2 4 6 7 2 8 2 103 7 2 7 4 2 4 4 2 6 8 5 4 4 8 4 7 10 9 10 9 104 8 2 8 5 4 8 6 5 7 8 3 9 7 8 8 5 6 7 10 7 105 9 5 8 6 3 4 4 3 4 4 2 1 2 4 1 4 1 4 3 2
Survey 3 Respondent Ratings:
R/C 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2.0 3.0 2.0 1.0 3.5 2.5 2.0 1.0 3.0 4.0 3.5 3.0 1.0 1.5 2.0 1.0 3.0 2.5 1.0 1.5 2 2.5 2.5 2.0 1.5 2.0 2.0 4.6 2.0 3.0 4.5 5.5 2.0 1.0 2.0 2.2 2.3 2.1 2.2 1.8 3.0 3 8.5 7.4 6.5 1.7 7.6 5.8 4.1 8.1 7.0 9.1 6.9 4.0 1.2 5.1 6.7 5.5 6.9 6.7 2.5 1.9 4 7.5 3.4 3.1 2.8 4.7 4.0 4.8 3.5 6.1 5.8 6.0 4.6 1.2 6.3 6.6 3.4 5.8 6.8 1.9 3.8 5 5.0 8.0 5.0 4.0 8.0 7.0 8.0 6.0 9.0 10.0 3.0 7.0 1.0 7.0 9.0 4.0 8.0 7.5 1.5 4.0 6 3.7 2.7 4.4 2.5 5.7 4.5 5.4 4.2 3.0 4.4 4.0 2.2 1.0 5.0 5.4 1.8 6.4 6.2 1.5 6.9 7 7.0 6.0 6.0 5.0 6.0 7.0 8.0 6.0 8.0 9.0 9.0 9.0 5.0 8.0 10.0 6.0 9.0 10.0 6.0 7.0 8 6.5 6.8 7.3 4.9 7.5 7.2 7.6 5.5 6.6 6.0 7.9 5.5 4.5 6.5 7.0 5.0 7.0 8.0 5.0 6.1 9 3.1 1.6 4.5 3.9 8.0 6.5 5.6 7.8 10.0 7.8 6.5 1.0 2.1 7.9 5.4 3.9 6.1 2.0 1.0 3.5 10 5.0 6.0 4.5 3.0 7.0 6.0 6.0 4.0 6.5 7.0 6.0 6.0 3.5 6.0 7.0 5.5 5.5 6.5 4.5 4.0
316
R/C 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 11 5.3 6.0 6.1 4.1 7.0 5.4 4.8 5.0 6.2 6.4 6.8 3.8 1.0 4.2 4.0 2.2 7.3 7.8 1.9 3.0 12 4.0 3.0 5.0 1.0 3.0 2.0 4.0 2.0 3.0 2.0 7.0 3.0 1.0 2.0 2.0 1.0 3.0 2.0 1.0 1.0 13 3.7 4.3 2.4 2.1 3.3 4.2 3.6 3.4 3.1 4.5 4.4 5.4 3.2 4.7 6.4 3.6 7.4 6.5 3.6 6.7 14 7.0 8.0 8.0 4.0 5.0 9.0 5.0 8.0 9.0 9.0 9.0 9.0 1.0 2.0 4.0 5.0 5.0 7.0 5.0 7.0 15 5.3 4.3 4.1 1.0 4.2 4.3 4.3 3.5 3.5 4.1 4.2 4.8 1.2 7.0 6.8 5.0 10.0 9.7 4.0 7.7 16 7.0 7.0 3.0 2.0 8.0 6.0 6.0 2.0 7.0 8.0 3.0 6.0 3.0 5.0 6.0 8.0 10.0 5.0 7.0 8.0 17 8.0 4.0 5.1 5.5 6.0 6.7 7.0 4.9 5.2 6.9 7.0 5.1 8.1 4.2 4.5 6.6 1.6 9.0 4.3 2.1 18 7.0 6.0 6.0 1.5 8.0 8.5 7.0 2.0 8.0 8.5 6.5 5.0 1.5 4.5 8.0 2.0 4.0 7.0 1.5 6.0 19 6.5 5.5 6.3 3.3 7.0 7.7 6.8 5.8 4.3 8.5 7.5 6.0 2.0 8.5 9.0 7.5 6.7 5.6 3.3 6.3 20 9.0 8.0 5.0 3.0 8.5 7.0 7.0 7.0 7.5 7.5 9.0 8.0 5.0 8.0 7.5 6.0 7.5 7.5 6.5 8.0 21 3.0 2.5 1.0 1.0 3.0 2.0 3.8 1.0 3.5 3.7 3.5 2.0 1.0 1.5 2.0 2.3 3.5 3.0 1.0 1.0 22 4.5 5.5 5.1 3.0 6.3 6.2 5.9 4.7 6.0 6.5 6.5 4.5 1.0 3.7 6.3 3.9 6.9 6.7 2.5 6.5 23 5.0 6.0 4.5 3.0 7.5 5.0 8.0 4.0 4.5 8.5 8.0 4.0 2.0 7.8 8.0 4.0 7.0 5.0 2.0 4.0 24 9.0 10.0 7.0 6.0 8.0 10.0 8.0 10.0 8.0 6.0 9.0 8.0 3.0 4.0 9.0 4.0 8.0 9.0 4.0 10.0 25 5.5 6.0 5.7 1.2 6.8 6.8 6.2 2.0 7.0 8.0 5.5 7.0 1.1 5.0 8.2 1.5 6.1 8.0 1.5 2.0 26 4.7 6.1 2.0 1.0 4.1 6.2 5.0 4.0 4.5 6.5 4.2 3.3 1.0 3.5 6.3 1.2 5.8 4.8 1.0 2.7 27 8.2 7.3 6.5 5.6 8.3 7.9 6.5 5.3 7.1 8.9 7.4 5.1 3.4 5.2 5.1 6.2 6.2 7.9 5.2 6.2 28 3.0 3.7 2.0 1.0 4.5 4.8 5.2 3.8 6.0 9.9 6.0 4.0 1.5 5.0 8.0 4.2 9.0 7.0 3.5 2.5 29 3.4 5.0 2.3 1.0 3.8 6.7 4.9 1.1 1.5 7.9 4.7 3.9 1.0 6.1 6.8 1.7 8.6 9.8 2.5 3.6 30 5.6 2.2 4.3 1.4 5.8 4.5 6.8 1.3 7.1 7.4 1.8 2.0 0.6 6.4 7.0 2.6 6.9 6.3 1.2 1.0 31 7.5 3.0 7.8 3.0 8.5 7.0 7.4 6.0 8.1 8.5 8.9 9.4 2.0 9.0 8.7 5.0 8.0 8.4 7.0 7.3 32 5.5 3.0 5.0 1.0 6.5 6.0 6.0 3.0 5.0 7.0 4.5 3.5 1.0 4.0 5.0 4.0 6.0 4.0 1.0 3.0 33 1.0 1.5 1.0 1.0 2.0 2.5 3.0 2.0 4.3 6.0 6.0 2.0 1.2 6.2 6.1 1.0 4.0 4.3 2.0 2.2 34 3.3 4.3 4.1 1.4 4.2 1.8 4.9 1.9 3.2 5.9 5.1 2.2 1.1 1.7 3.4 1.2 3.9 2.1 1.2 2.2 35 7.5 4.7 6.5 3.5 6.0 5.8 7.2 6.0 6.8 7.0 6.9 4.5 2.5 7.5 6.8 4.5 8.3 8.3 4.2 5.3 36 5.8 7.8 6.3 1.7 6.8 7.3 8.2 7.0 7.7 8.0 6.3 6.8 1.5 8.3 7.5 2.5 8.5 6.7 1.5 6.5 37 8.0 9.0 7.0 5.0 7.0 9.0 8.0 9.0 8.0 6.0 10.0 8.0 3.0 6.0 5.0 6.0 6.0 8.0 2.0 6.0 38 6.5 7.0 5.5 3.5 8.5 7.5 8.5 7.5 8.5 9.0 8.0 5.0 1.0 5.5 8.0 3.5 4.0 9.5 3.0 6.0 39 5.5 5.7 4.0 3.1 5.5 4.8 6.1 3.5 4.6 6.1 5.0 3.9 1.0 5.3 3.9 4.3 5.6 6.0 3.0 3.9 40 3.5 1.5 4.0 2.0 6.0 6.5 9.5 7.5 9.0 6.5 7.0 5.0 1.5 7.0 6.5 4.5 10.0 8.0 4.0 6.0 41 6.0 4.0 4.0 1.0 5.0 4.0 4.0 1.0 3.0 4.0 3.0 2.0 4.0 4.0 4.0 2.0 5.0 5.0 3.0 2.0 42 3.5 5.5 4.8 1.1 6.5 5.8 5.7 2.8 3.5 4.5 3.8 5.7 0.5 1.8 4.3 0.7 2.3 4.5 0.6 3.8 43 7.0 6.0 3.0 1.0 5.0 5.0 3.0 4.0 4.0 8.0 6.0 7.0 1.0 3.0 9.0 2.0 3.0 7.0 2.0 2.0 44 2.0 7.0 7.0 2.0 4.0 4.0 5.0 3.0 3.5 2.5 5.5 4.6 1.0 3.9 6.0 7.1 7.2 6.1 1.1 4.2 45 7.5 6.0 8.0 6.0 7.0 8.0 9.5 6.5 8.5 7.5 6.5 7.5 4.0 5.0 5.5 4.0 6.0 7.0 4.0 5.0 46 6.0 7.0 4.0 2.0 6.0 8.0 7.0 2.0 4.0 8.0 3.0 4.0 1.0 8.0 9.0 6.0 7.0 10.0 3.0 4.0 47 5.0 6.0 4.0 4.0 5.0 5.0 4.0 5.0 5.0 7.0 5.0 6.0 1.0 7.0 5.0 5.0 7.0 6.0 3.0 3.0 48 2.1 8.5 5.3 2.0 6.0 7.0 6.0 3.0 7.0 7.9 7.5 4.0 1.0 4.0 6.0 2.0 6.0 6.0 2.5 5.5 49 5.5 6.2 4.7 3.6 6.9 5.4 7.9 4.0 6.5 6.3 7.5 4.7 1.3 7.3 5.7 3.4 8.9 7.3 2.1 5.0 50 5.4 4.3 5.3 3.7 4.4 5.3 6.4 3.4 6.3 6.6 6.6 5.4 2.3 3.4 4.2 3.5 4.8 6.6 4.4 6.5 51 1.0 1.5 2.0 1.0 2.5 2.0 2.0 1.0 1.5 1.5 3.0 1.0 1.0 1.0 2.0 1.0 2.5 1.5 1.0 1.0 52 7.0 7.0 5.0 2.0 8.0 8.0 8.5 6.0 7.5 7.8 6.5 3.0 1.0 4.0 6.8 1.5 7.0 6.2 2.5 5.0 53 5.0 10.0 3.0 2.0 9.0 6.0 7.0 4.0 7.0 10.0 7.0 7.0 4.0 6.0 7.0 6.0 7.0 6.0 4.0 5.0 54 6.8 5.4 2.3 1.0 7.8 6.5 6.0 2.5 2.0 5.2 8.0 4.4 1.0 4.6 7.0 2.9 8.2 3.7 1.0 3.5 55 3.5 3.3 5.5 1.0 3.5 3.4 5.8 5.2 5.8 6.7 7.2 1.0 1.0 1.5 4.7 1.0 6.8 2.4 1.5 2.4 56 7.7 9.0 6.3 4.1 6.3 3.5 7.6 6.5 5.2 7.5 7.5 6.0 1.5 7.0 8.0 7.0 8.7 7.3 5.0 5.3 57 4.0 7.5 4.5 4.5 6.5 4.5 6.5 6.0 5.5 5.5 7.0 5.5 3.0 6.0 7.0 3.5 5.5 5.0 4.5 5.0 58 5.2 7.3 6.7 5.2 6.2 6.4 7.1 7.9 6.4 7.6 7.1 7.8 2.3 7.9 8.5 3.6 8.6 8.3 3.6 5.2 59 3.0 4.0 2.0 1.0 4.0 4.0 5.0 4.0 6.0 7.0 5.0 3.0 1.0 4.0 5.0 4.0 5.0 6.0 1.0 4.0 60 4.0 5.0 3.5 1.0 6.5 5.5 5.5 4.5 6.0 7.0 4.0 3.5 1.0 6.5 7.5 1.5 8.0 2.0 1.0 1.0 61 2.2 1.3 3.3 5.4 4.6 3.5 6.6 2.1 8.1 5.2 6.7 2.0 1.0 3.5 7.8 6.3 5.3 7.3 4.2 4.5 62 6.0 7.0 4.0 1.0 4.0 4.0 4.5 3.0 5.5 7.0 4.0 3.0 1.0 5.0 6.0 2.0 5.0 7.0 1.0 1.0 63 5.0 9.0 2.0 3.0 8.0 4.0 5.2 1.5 8.0 6.5 8.5 1.5 1.0 2.5 9.5 1.2 8.8 9.7 8.3 2.5
317
R/C 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 64 5.0 9.0 5.0 1.0 3.0 5.0 7.0 5.0 3.0 10.0 6.0 6.0 1.0 4.0 3.0 2.0 5.0 5.0 1.0 5.0 65 5.0 5.0 5.0 4.0 6.0 5.0 5.0 5.0 5.0 6.0 7.0 4.0 4.0 5.0 8.0 5.0 7.0 7.0 5.0 5.0 66 5.0 7.0 1.5 0.5 7.5 7.0 8.5 6.5 7.5 7.0 7.0 4.5 0.5 3.5 4.5 0.1 6.5 6.0 2.0 6.5 67 4.0 4.0 5.0 3.0 4.0 6.0 8.0 7.0 7.0 7.0 8.0 7.0 2.0 4.0 4.0 2.0 5.0 7.0 2.0 3.0 68 3.0 4.0 2.0 2.0 4.5 3.2 5.0 3.8 5.1 5.2 3.0 2.8 1.2 2.3 2.5 1.5 3.6 5.2 1.8 2.1 69 7.0 5.0 4.0 1.0 6.0 5.5 4.5 3.0 3.5 5.7 4.5 5.2 1.0 2.5 2.4 2.0 5.7 5.3 1.5 1.6 70 8.0 9.5 3.0 2.0 7.0 8.0 8.5 2.0 9.0 5.0 10.0 8.5 1.0 6.0 9.5 1.0 7.0 10.0 2.0 4.0 71 5.0 4.5 3.5 3.5 6.0 6.0 7.0 5.0 7.0 7.0 6.0 4.0 1.0 3.5 4.0 2.5 4.0 5.5 2.0 2.5 72 9.0 9.5 1.5 3.0 6.0 5.0 8.0 2.0 4.0 7.0 9.0 1.0 1.0 5.0 8.0 3.0 9.5 7.0 2.0 10.0 73 9.0 6.0 5.0 4.0 9.0 5.0 8.0 4.0 7.5 9.5 7.5 8.5 1.0 6.5 9.0 4.5 10.0 8.5 3.0 7.5 74 5.5 7.7 4.5 3.5 7.5 6.0 8.0 7.2 8.7 7.9 7.2 4.5 2.2 6.4 9.0 3.5 8.9 7.2 3.3 8.5 75 6.8 6.5 3.0 3.5 8.0 8.2 7.3 6.0 7.0 8.0 6.7 5.0 1.5 5.4 7.2 2.0 2.1 4.0 2.5 4.6 76 8.0 7.5 7.6 7.7 7.9 8.5 8.2 7.7 7.8 8.6 9.0 8.4 6.0 9.0 9.1 7.5 9.0 8.8 8.0 8.7 77 3.0 7.0 2.0 1.0 7.0 6.0 6.5 4.0 6.0 9.0 6.0 3.0 1.0 7.0 8.0 4.0 9.0 7.0 1.1 2.0 78 0.3 0.5 0.2 0.1 0.3 0.3 0.6 0.3 0.3 0.4 0.3 0.3 0.1 0.2 0.3 0.1 0.2 0.3 0.1 0.3 79 6.5 7.5 6.0 7.0 8.5 4.5 9.0 10.0 3.5 4.5 4.0 5.0 5.5 7.0 6.5 5.5 5.5 10.0 7.5 9.5 80 5.0 5.0 4.0 3.0 3.0 4.5 2.5 3.5 2.0 5.0 3.5 5.5 5.0 4.5 6.5 6.0 5.0 6.0 5.0 8.0 81 2.2 7.8 6.0 3.5 7.0 7.1 8.0 2.5 3.2 6.4 7.3 6.9 0.8 7.4 7.7 5.9 7.6 6.5 1.3 2.6 82 5.8 4.7 3.8 3.1 5.9 5.3 5.7 4.1 4.3 4.6 4.0 2.8 2.4 2.6 2.6 2.3 5.7 5.5 2.7 2.8 83 8.9 9.5 7.5 1.5 8.0 9.6 8.0 7.0 6.0 9.0 10.0 8.0 1.0 9.9 10.0 1.2 9.7 9.7 1.2 9.6 84 8.0 7.5 5.9 6.0 6.5 6.5 6.0 7.0 7.5 8.0 8.0 8.2 6.0 6.5 8.3 6.5 7.7 8.7 5.9 7.2 85 5.0 5.0 5.0 4.0 4.0 4.0 6.0 6.0 5.0 4.0 4.0 5.0 3.0 4.0 4.0 3.0 6.0 7.0 4.0 4.0 86 7.0 7.5 5.0 4.5 6.0 6.0 6.5 5.5 6.0 5.5 7.5 5.0 5.0 7.0 6.0 5.5 6.5 6.5 4.5 5.0 87 5.0 7.0 3.0 1.0 6.0 7.0 6.0 2.0 6.0 4.0 6.0 2.0 1.0 6.0 6.0 3.0 6.0 7.0 1.0 5.0 88 8.0 5.5 7.5 5.0 7.5 7.0 6.5 7.0 8.0 8.5 7.5 8.0 5.0 7.5 8.5 7.0 8.0 8.0 7.0 6.0 89 5.0 5.0 4.0 3.0 7.0 6.0 6.0 4.0 3.0 7.0 7.0 5.0 3.0 8.0 5.0 6.0 2.0 7.0 8.0 4.0 90 6.5 5.0 4.0 1.0 7.0 6.0 6.5 4.0 5.5 6.0 7.0 5.0 3.0 5.5 5.0 2.0 7.0 5.5 1.0 3.0 91 5.0 5.0 4.0 3.0 6.0 6.0 5.0 4.0 5.0 5.0 6.0 5.0 1.0 6.0 6.0 2.0 7.0 6.0 3.0 4.0 92 3.0 7.5 2.0 1.1 6.0 6.0 7.0 1.2 2.0 8.0 8.0 5.0 1.0 2.5 1.5 1.1 8.5 2.0 1.0 2.0 93 6.5 7.0 5.0 1.0 8.0 4.0 9.0 2.0 4.5 8.5 5.5 3.5 1.2 1.5 7.5 1.3 8.3 6.5 1.1 3.0 94 8.0 8.0 8.0 3.0 8.0 8.0 8.0 4.0 5.0 6.0 7.0 6.0 1.0 4.5 7.0 2.0 5.5 6.5 2.0 4.0 95 7.0 7.0 4.6 4.0 6.0 7.0 4.0 7.0 8.0 6.0 5.0 5.0 1.0 5.3 5.0 2.0 5.0 7.5 3.0 3.0 96 8.8 7.0 5.3 3.5 8.9 8.0 7.8 6.5 6.8 9.7 8.7 9.1 1.0 5.7 5.5 3.0 9.8 9.4 3.1 3.2 97 6.0 4.5 4.0 3.0 6.0 4.5 6.0 3.0 6.0 6.5 6.2 6.0 2.0 4.5 4.5 3.0 5.5 4.5 3.0 3.0 98 2.5 2.0 3.5 1.0 3.5 3.0 4.0 1.4 4.0 5.0 3.0 3.0 1.0 1.6 2.7 1.3 4.0 2.8 1.0 2.0 99 5.0 7.0 6.0 5.0 5.5 4.0 7.0 7.0 7.0 8.0 7.5 7.0 2.0 7.5 7.5 7.0 8.0 7.0 5.0 7.5
100 4.0 6.0 3.1 2.5 4.6 3.9 4.8 3.2 6.4 7.0 6.1 2.9 1.0 2.8 5.7 2.2 6.8 4.4 1.6 4.0 101 5.6 8.5 7.9 5.4 5.2 8.7 9.7 4.1 4.2 8.0 7.1 7.6 3.5 7.9 7.2 6.7 7.7 7.9 6.4 6.4 102 7.7 5.1 3.1 7.1 5.1 7.6 8.2 3.0 4.5 8.3 9.4 8.0 1.0 8.9 9.3 2.7 7.5 9.1 1.2 4.5
Survey 4 Respondent Ratings:
R/C 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 4.8 5.6 7.2 7.2 8.6 3.6 6.3 8.0 7.8 5.4 3.2 9.1 4.5 8.9 8.3 6.0 8.5 4.2 6.1 8.7 2 6.0 3.0 2.5 2.6 8.0 5.0 4.0 5.0 3.0 3.0 4.5 4.0 2.4 2.3 2.4 3.0 3.1 1.5 4.0 7.5 3 5.0 5.5 6.8 7.6 9.0 7.4 3.0 4.5 8.0 3.5 6.7 8.2 6.3 10.0 8.5 4.6 6.5 7.0 7.6 8.6 4 5.2 5.5 6.0 5.4 7.5 5.5 8.0 4.0 6.5 4.5 7.9 6.0 3.4 8.5 5.0 6.6 5.5 6.0 6.9 7.5 5 2.2 5.3 5.9 6.3 8.1 7.9 7.9 1.5 5.0 1.7 1.6 6.0 3.0 5.8 4.8 3.5 3.9 2.5 4.9 5.5 6 3.2 5.5 4.1 3.5 5.5 7.5 3.1 4.5 4.0 2.0 3.4 5.4 2.1 3.5 4.2 6.5 4.1 1.9 3.0 4.0 7 7.0 6.5 5.5 5.0 8.0 6.0 5.0 4.0 3.5 3.0 4.5 5.0 3.0 4.0 5.0 4.0 4.0 2.5 4.0 4.0 8 1.0 1.5 4.5 3.0 5.0 2.0 3.5 2.0 1.0 1.0 1.0 5.0 2.0 5.5 4.5 2.0 2.5 1.0 1.5 2.5 9 4.0 7.9 4.5 6.0 9.0 3.5 4.1 6.5 7.0 7.2 3.5 3.0 6.4 5.0 3.5 4.9 6.7 5.9 8.1 8.5
318
R/C 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 10 2.5 2.2 6.1 5.2 7.4 6.8 3.3 1.7 2.8 3.1 2.1 8.0 1.7 3.4 2.9 1.4 4.2 2.0 3.5 6.3 11 2.4 2.1 3.2 4.5 5.9 2.0 3.3 1.0 3.0 4.0 0.1 1.1 1.9 0.0 0.0 5.5 5.0 1.0 1.2 3.8 12 8.0 6.5 6.0 7.5 8.0 6.5 6.0 6.0 7.5 6.0 5.5 6.0 5.5 7.0 7.0 5.0 6.5 5.5 7.5 5.0 13 6.3 6.6 7.1 7.3 7.5 6.7 6.5 6.0 6.5 6.8 6.6 7.8 6.5 7.1 6.8 6.6 6.7 7.2 7.4 6.9 14 6.5 5.0 5.5 5.6 6.1 5.3 5.4 5.0 6.0 5.0 3.5 3.5 5.4 4.9 6.0 5.3 5.5 5.5 6.1 6.0 15 7.0 1.0 6.0 5.0 4.5 9.0 8.0 1.5 10.0 8.5 7.0 7.5 8.0 7.0 3.5 9.0 4.5 6.0 5.5 7.0 16 1.5 1.5 1.5 1.5 4.0 1.0 3.0 1.0 1.5 1.0 1.0 3.0 1.0 1.5 1.0 4.0 1.5 1.0 1.0 1.0 17 2.0 4.0 1.0 1.0 9.0 7.0 2.0 4.0 1.0 3.0 1.0 6.5 3.0 1.0 1.0 6.5 5.0 3.5 2.0 2.0 18 5.8 3.0 3.5 7.5 8.0 7.0 7.5 5.5 8.5 5.0 8.0 8.5 4.0 7.5 8.2 6.0 5.5 8.6 8.0 7.0 19 3.9 4.6 3.6 5.5 8.1 3.1 6.5 2.9 6.8 3.2 6.7 6.1 3.7 6.9 5.9 5.0 5.7 4.2 7.8 8.5 20 2.0 3.0 2.0 5.0 7.0 2.0 7.0 2.0 2.0 7.0 3.0 3.0 2.0 7.0 4.0 1.0 1.0 2.0 8.0 2.0 21 6.2 7.8 4.5 6.5 7.6 5.5 4.2 7.2 2.1 3.1 8.1 8.5 4.5 5.1 6.5 4.5 8.1 4.5 7.5 7.5 22 1.0 1.0 5.0 7.0 7.0 1.0 5.0 1.0 5.0 5.0 5.0 5.0 3.0 5.0 4.0 4.0 4.0 1.0 5.0 5.0 23 1.0 1.5 6.5 4.0 4.5 5.5 4.0 1.0 5.5 3.0 2.0 6.0 3.0 7.0 4.0 3.5 5.0 5.5 7.5 6.0 24 3.0 4.0 5.0 7.0 4.0 3.0 6.0 5.0 4.0 3.0 6.0 5.0 5.0 5.0 3.0 6.0 5.0 2.0 5.0 4.0 25 8.0 7.0 4.0 5.0 8.0 5.0 5.0 2.0 6.0 4.0 3.0 8.0 3.0 8.0 4.0 6.0 3.0 2.0 4.0 3.0 26 6.7 4.3 7.6 8.5 8.3 5.6 8.1 2.7 4.3 1.3 3.5 7.9 5.9 6.8 9.1 5.6 7.3 7.2 7.6 6.3 27 2.4 3.3 5.6 6.1 4.1 3.3 5.2 2.6 2.1 4.0 3.7 6.6 5.4 6.0 7.6 4.6 6.4 8.1 6.8 5.9 28 2.5 3.0 4.4 6.3 8.3 5.5 9.2 5.2 4.5 9.3 6.8 7.3 4.7 3.4 8.2 5.6 6.1 8.6 9.2 7.2 29 8.0 6.0 4.0 3.0 7.0 7.0 6.0 4.0 5.0 7.0 7.5 3.0 6.0 6.5 8.5 7.5 6.0 5.0 4.5 4.5 30 6.5 4.0 3.0 7.0 8.0 9.6 7.6 1.0 6.0 3.0 5.5 4.5 3.4 7.8 5.2 6.1 2.4 2.3 4.5 2.5 31 7.5 3.2 7.2 7.8 8.0 2.0 6.8 7.4 4.5 7.5 2.5 8.5 4.0 7.2 3.1 2.0 6.5 5.0 7.0 7.6 32 8.5 8.0 7.1 8.2 9.9 6.5 6.0 7.0 5.5 7.9 5.5 5.2 6.2 6.1 7.1 7.8 6.9 4.4 8.1 9.5 33 10.0 10.0 9.0 4.0 10.0 10.0 6.0 8.0 8.0 9.0 9.0 10.0 8.0 10.0 10.0 7.0 1.0 7.0 7.0 5.0 34 8.0 3.0 5.0 6.0 8.0 2.0 6.0 2.0 8.0 4.0 6.0 5.0 3.0 6.0 4.0 5.0 6.0 4.0 3.0 3.0 35 3.0 2.1 2.4 2.5 8.0 6.0 2.0 1.0 2.0 2.7 2.6 2.0 2.3 2.0 2.2 2.0 1.5 2.5 3.0 3.5 36 6.8 7.0 5.9 7.5 7.4 6.0 7.2 3.0 6.9 7.6 8.0 7.4 7.2 7.0 7.5 5.4 7.6 6.7 7.0 5.0 37 8.3 7.7 8.5 6.2 9.2 5.7 8.5 4.7 9.4 6.8 7.7 9.7 1.6 5.9 5.7 3.8 2.8 2.3 5.5 4.8 38 6.0 5.0 8.0 8.0 8.5 5.0 4.0 3.0 5.0 3.0 2.5 2.5 2.0 4.0 6.0 2.0 3.0 3.0 5.0 7.0 39 6.8 4.0 5.5 7.3 7.2 2.0 3.5 4.2 3.0 7.5 4.5 5.2 6.3 5.9 6.5 8.5 6.0 3.2 7.1 4.1 40 6.1 5.7 4.9 5.5 8.0 6.5 7.0 4.1 7.5 6.5 5.0 6.0 5.2 7.2 7.0 6.0 5.8 6.1 6.0 7.0 41 4.9 4.7 5.0 7.8 7.5 3.1 4.5 3.8 7.2 3.3 3.5 8.2 2.5 8.8 6.3 6.9 5.3 2.3 2.5 8.5 42 4.2 6.1 3.7 3.5 7.0 4.9 6.6 1.8 4.6 3.9 7.7 8.7 2.2 8.3 4.5 9.4 6.4 3.3 5.7 6.5 43 7.0 1.5 1.0 5.0 10.0 10.0 8.0 5.0 6.0 6.0 6.5 10.0 1.0 7.5 8.5 2.0 3.5 5.0 9.0 10.0 44 3.0 2.0 2.0 2.0 7.0 5.0 5.0 4.0 7.0 8.0 5.0 8.0 1.0 6.0 6.0 8.0 5.0 2.0 5.0 4.0 45 3.5 2.8 5.5 6.0 6.0 9.0 6.5 4.0 9.2 3.9 4.3 8.7 5.0 8.9 7.4 7.2 6.5 3.1 5.8 10.0 46 1.7 1.2 7.2 4.9 8.9 5.2 3.7 4.0 2.6 3.3 1.8 7.3 2.0 4.6 5.7 9.7 6.8 1.4 6.3 2.4 47 9.0 5.0 5.0 9.0 10.0 8.0 5.0 5.0 6.0 8.5 8.0 9.5 5.0 8.0 9.0 9.0 8.0 3.0 7.0 8.0 48 6.0 9.7 5.5 4.3 10.0 7.8 4.5 8.0 7.7 5.7 9.5 6.3 8.7 9.7 5.6 3.1 6.1 5.9 6.8 2.1 49 3.0 4.0 7.5 5.0 7.5 6.0 5.5 3.0 7.0 6.0 8.5 7.0 3.5 8.2 9.0 9.0 5.5 6.0 7.5 7.5 50 3.0 2.0 8.5 9.0 10.0 2.0 7.5 1.0 5.3 6.5 8.9 9.4 6.5 3.2 1.5 4.7 6.9 9.5 8.4 3.2 51 3.0 5.0 6.0 4.0 3.0 6.0 6.0 7.0 4.0 6.0 4.0 3.0 5.0 4.0 4.0 4.0 3.0 4.0 5.0 5.0 52 5.0 7.0 6.0 7.0 8.0 7.0 7.0 6.0 6.5 6.0 7.0 7.5 6.0 6.8 7.0 6.0 6.5 6.5 7.0 7.6 53 7.0 8.5 8.0 6.5 8.5 1.0 2.0 8.0 3.0 2.0 5.0 6.0 3.5 6.5 5.5 4.5 7.0 5.5 7.5 2.5 54 3.5 2.0 6.5 3.0 7.0 4.4 8.7 8.1 7.2 4.0 5.0 9.0 4.5 7.6 6.6 8.2 8.5 3.0 8.9 7.9 55 3.4 3.1 4.5 5.3 7.8 4.2 4.5 3.5 3.7 5.2 4.8 4.1 4.4 6.1 5.8 5.9 4.9 3.8 6.5 7.2 56 2.0 6.2 8.0 7.5 9.2 3.0 5.5 3.0 9.5 5.0 6.5 7.0 8.5 3.0 8.0 6.0 7.0 5.0 9.0 6.0 57 10.0 1.0 7.0 3.6 9.0 2.0 2.0 1.5 10.0 6.0 8.0 5.0 1.0 7.0 6.0 2.5 3.0 1.0 3.0 7.0 58 5.1 7.2 6.3 4.8 7.9 2.4 3.8 8.7 5.3 6.4 4.4 8.3 3.2 2.5 4.6 8.1 8.9 4.9 4.9 5.2 59 6.4 3.2 3.1 7.6 8.8 5.3 8.9 3.0 7.2 7.8 8.5 8.2 5.0 7.9 8.0 8.1 8.6 6.0 9.0 7.2 60 1.0 2.0 6.0 7.0 8.0 8.0 7.0 9.0 2.0 5.0 6.0 5.0 7.0 8.0 4.0 5.0 9.0 7.0 5.0 8.0 61 5.0 4.0 6.0 8.0 9.0 4.0 5.0 4.0 1.0 1.0 2.0 8.0 4.0 2.0 2.0 3.0 7.0 2.0 7.5 6.0 62 3.3 1.2 3.3 6.0 7.0 7.1 2.0 7.5 2.0 7.3 1.0 0.9 0.8 0.7 4.0 5.0 8.0 5.0 7.0 5.0
319
R/C 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 63 7.0 5.0 7.5 6.0 8.0 2.0 4.0 7.5 7.0 9.0 6.0 9.5 4.0 4.0 4.0 2.0 7.0 4.0 5.0 4.0 64 3.0 3.5 4.0 5.3 4.5 5.0 5.5 4.7 5.0 6.0 5.6 5.1 5.4 5.9 6.5 6.4 6.1 5.9 5.5 6.1 65 3.0 2.0 3.0 5.0 9.0 1.0 1.0 2.0 2.0 2.0 4.0 1.0 2.0 1.0 2.0 3.0 4.0 2.0 2.0 3.0 66 1.0 2.0 1.0 3.0 2.0 3.0 1.0 1.5 1.9 2.1 1.0 1.5 1.0 2.0 2.0 1.5 1.8 1.1 1.2 1.3 67 1.0 2.0 2.5 2.0 8.0 2.0 1.0 1.0 1.0 1.0 1.5 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.5 68 6.0 6.0 10.0 10.0 7.0 9.0 7.5 8.0 5.0 6.0 8.0 8.0 4.5 8.5 10.0 6.0 10.0 4.0 6.5 4.0 69 5.0 4.0 7.0 4.0 4.0 6.0 8.0 9.0 7.0 4.0 6.0 7.0 7.0 4.0 7.0 2.0 5.0 5.0 9.0 9.0 70 8.7 1.0 4.0 6.0 9.0 8.5 7.8 3.6 8.2 2.0 5.4 9.7 5.8 9.5 7.9 7.3 5.7 8.8 4.7 9.4 71 6.5 4.0 7.5 5.5 8.0 7.0 7.5 3.5 6.0 9.0 9.5 8.5 5.0 8.0 6.5 7.0 6.0 4.5 9.5 8.5 72 3.8 4.0 3.0 3.0 8.0 2.5 3.4 1.1 3.4 2.5 2.0 5.8 2.5 3.4 3.0 1.0 2.5 3.0 3.0 7.5 73 7.0 7.0 7.0 7.0 9.5 4.0 7.0 7.0 8.0 6.0 4.0 8.0 5.0 7.0 7.5 8.0 8.5 8.0 8.0 8.0 74 9.4 4.8 5.0 7.3 9.5 8.0 3.3 3.2 3.0 4.1 4.2 7.5 2.8 5.0 6.9 2.3 4.0 1.2 3.7 6.7 75 7.0 2.0 2.0 4.0 7.0 5.0 5.0 7.0 6.0 7.0 2.0 9.0 2.0 7.0 7.0 8.0 7.0 1.0 5.0 6.0 76 4.0 5.2 6.7 6.7 8.1 8.1 8.5 6.1 8.9 8.1 9.5 9.5 7.1 8.0 8.5 6.5 5.1 7.9 7.9 8.0 77 4.1 7.9 3.3 6.4 9.0 9.1 8.3 3.2 7.5 6.6 2.9 9.0 2.8 7.5 5.7 4.6 5.1 3.0 4.4 6.9 78 2.3 3.2 4.2 5.8 8.0 3.0 6.6 3.5 1.5 2.5 1.3 1.9 2.0 3.1 4.4 1.2 9.0 1.8 2.9 2.1 79 7.5 8.0 9.0 10.0 7.5 9.5 4.0 6.5 8.5 9.5 4.2 5.5 5.6 7.8 9.0 4.0 5.0 5.5 6.5 7.8 80 1.0 3.0 7.0 2.0 7.0 3.0 5.0 1.0 3.0 4.0 4.0 2.0 1.0 3.0 2.0 1.0 2.0 1.0 3.0 5.0 81 6.0 4.8 7.5 8.0 7.0 6.0 5.0 3.0 5.0 4.0 7.0 4.0 3.0 7.5 7.0 7.9 8.0 4.0 5.0 6.0
1.6 Screen Captures
The following (Figure F.9) is a screen capture of survey 3. The basic structure was the
same for all the surveys. Respondents could interactively select and play through the
combinations to the left of the screen and then rate them in the corresponding boxes
provided on the right. The control questions were placed in the frame to the right, after
the instruction set and respondent information fields but before the commentary box and
submission button (Figure F.10). Once all the ratings for the 20 combinations had been
entered and the necessary fields completed (including answers to the control questions),
respondents simply clicked the ‘submit’ button at the very bottom of the page. The
results were automatically e-mailed to the author. A simple computer program was
written to parse these e-mails into a spreadsheet-friendly format where the data was
organized and later analyzed. The survey was available online and running for
approximately one month between June and July 2008.
320
Figure F.9 Survey 3: Screen Capture 1
Figure F.10 Survey 3: Screen Capture 2
321
SELECTED PUBLICATIONS
International Journals 1. Iqbal, A. and Yaacob, M. (2008). Theme Detection and Evaluation in Chess, ICGA
Journal, Vol. 32, No. 2, pp. 97-109. 2. Iqbal, A. (2008). Evaluation of Economy in a Zero-sum Perfect Information Game,
The Computer Journal, Oxford University Press, Vol. 51, No. 4, pp. 408-418. 3. Iqbal, A. (2006). Is Aesthetics Computable? ICGA Journal, Vol. 29, No. 1, pp. 32-
39. 4. Iqbal, A. and Yaacob, M. (2006). A Systematic and Discrete View of Aesthetics in
Chess, Journal of Comparative Literature and Aesthetics, Vol. XXIX, Nos. 1-2, pp. 53-65; ISSN: 0252-8169.
5. Iqbal, A. and Yaacob, M. (2005). Computational Aesthetics and Chess as an Art
Form, Journal of Comparative Literature and Aesthetics, Vol. XXVIII, Nos. 1-2, pp. 49-59; ISSN: 0252-8169.
International Conferences 1. Iqbal, A. and Yaacob, M. (2008). Computational Assessment of Sparsity in Board
Games, Proceedings of the 12th International Conference on Computer Games: AI, Animation, Mobile, Educational and Serious Games (CGames 2008), Louisville, Kentucky, USA, 30 July - 2 August, pp. 29-33.
2. Iqbal, A. (2006). Computing the Aesthetics of Chess, Technical Report of the AAAI
'06 Workshop on Computational Aesthetics (WS-06-04), Organized by Massachusetts Institute of Technology and University of North Texas, AAAI Press, Boston, Massachusetts, USA, 16-20 July, pp. 16-23.